Abstract

We study focused fields which, for a given total power and a given numerical aperture, have maximum electric field amplitude in some direction in the focal point and are linearly polarized along this direction. It is shown that the optimum field is identical to the image of an electric dipole with unit magnification. In particular, the field which is the image of an electric dipole whose dipole vector is parallel to the optical axis, is identical to the field whose longitudinal component is maximum at the image point.

© 2011 OSA

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  1. V. S. Ignatowsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd I, paper IV (1919), (in Russian).
  2. V. S. Ignatowsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans. Opt. Inst. Petrograd , I, paper V (1920), (in Russian).
  3. E. Wolf, “Electromagnetic diffraction in optical systems I. An integral repesentation of the image field,” Proc. R. Soc. London, Ser. A 253, 349–357 (1959).
    [CrossRef]
  4. 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. A 253, 358–379 (1959).
    [CrossRef]
  5. X. S. Xie and R. C. Dunn, “Probing single molecule dynamics,” Science 265, 361–364 (1994).
    [CrossRef] [PubMed]
  6. L. Novotny, M. R. Beverluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
    [CrossRef] [PubMed]
  7. Q. W. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004).
    [CrossRef] [PubMed]
  8. L. E. Helseth, “Focusing of atoms with strongly confined light potentials,” Opt. Commun. 212, 343–352 (2002).
    [CrossRef]
  9. M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A: Mater. Sci. Process. 86, 329–334 (2007).
    [CrossRef]
  10. Q. Zhan and J. Leger, “Focus Shaping using cylindrical vector beams,” Opt. Express 10, 324–331 (2002).
    [PubMed]
  11. J. Wang, W. Chen, and Q. Zhan, “Engineering of high purity ultra-long optical needle field through reversing the electric dipole array radiation,” Opt. Express 18, 21965–21972 (2010).
    [CrossRef] [PubMed]
  12. N. Sanner, N. Huot, E. Audouard, C. Larat, J.-P. Huignard, and B. Loiseaux, “Programmable focal spot shaping of amplified femtosecond laser pulses,” Opt. Lett. 30, 1479–1481 (2005).
    [CrossRef] [PubMed]
  13. M. A. A. Neil, F. Massoumian, R. Juskaitis, and T. Wilson, “Method for the generation of arbitrary complex vector wave fronts,” Opt. Lett. 27, 1929–1931 (2002).
    [CrossRef]
  14. M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21, 1948 (1996).
    [CrossRef] [PubMed]
  15. I. Iglesias and B. Vohnsen, “Polarization structuring for focal volume shaping in high-resolution microscopy,” Opt. Commun. 271, 40–47 (2007).
    [CrossRef]
  16. C. J. R. Sheppard and K. G. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41, 1495–1505 (1994).
    [CrossRef]
  17. C. J. R. Sheppard and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik 104, 175–177 (1997).
  18. A. J. E. M. Janssen, S. van Haver, J. J. M. Braat, and P. Dirksen, “Strehl ratio and optimum focus of high-numerical-aperture beams,” J. Eur. Opt. Soc. Rapid Publ. 2, 07008 (2007).
    [CrossRef]
  19. V. Dhayalan and J. J. Stamnes, “Focusing of mixed-dipole waves,” Pure Appl. Opt. 6, 317–345 (1997).
    [CrossRef]
  20. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
    [CrossRef] [PubMed]
  21. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
    [CrossRef]
  22. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light–theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
  23. C. J. R. Sheppard and A. Choudhurry, “Annular pupils, radial polarization and superresolution,” Appl. Opt. 43, 4322–4327 (2004).
    [CrossRef] [PubMed]
  24. H. P. Urbach and S. F. Pereira, “The field in focus with maximum longitudinal electric component,” Phys. Rev. Lett. 100, 1233904 (2008).
    [CrossRef]
  25. H. P. Urbach and S. F. Pereira, “Focused fields of given power with maximum electric field components,” Phys. Rev. A 79, 013825 (2009).
    [CrossRef]
  26. H. P. Urbach and S. F. Pereira, “Erratum: focused fields of given power with maximum electric field components,” Phys. Rev. A 81, 059903 (2010).
    [CrossRef]
  27. W. Chen and Q. Zhan, “Diffraction limited focusing with controllable arbitrary three-dimensional polarization,” J. Opt. 12, 045707 (2010)
    [CrossRef]
  28. E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
    [CrossRef]
  29. J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in “Progress in Optics ,” Vol. 51, E. Wolf, ed. (Elsevier B.V., 2008), chap. 6, pp. 349–468.
    [CrossRef]
  30. R. Aarts, J. J. M. Braat, P. Dirksen, S. van Haver, C. van Heesch, and A. Janssen, “Analytic expressions and approximations for the on-axis, aberration-free Rayleigh and Debye integral in the case of focusing fields on a circular aperture,” J. Eur. Opt. Soc. Rapid Publ. 3, 08039 (2008).
    [CrossRef]
  31. J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and S. F. Pereira, “Image formation in a multilayer using the extended Nijboer-Zernike theory,” J. Eur. Opt. Soc. Rapid Publ. 4, 09048 (2009).
    [CrossRef]
  32. L. Novotny and B. Hecht, “Principles of nano-optics,” Section 2.10.2, (Cambridge University Press, 2008).

2010

J. Wang, W. Chen, and Q. Zhan, “Engineering of high purity ultra-long optical needle field through reversing the electric dipole array radiation,” Opt. Express 18, 21965–21972 (2010).
[CrossRef] [PubMed]

H. P. Urbach and S. F. Pereira, “Erratum: focused fields of given power with maximum electric field components,” Phys. Rev. A 81, 059903 (2010).
[CrossRef]

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

2009

H. P. Urbach and S. F. Pereira, “Focused fields of given power with maximum electric field components,” Phys. Rev. A 79, 013825 (2009).
[CrossRef]

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and S. F. Pereira, “Image formation in a multilayer using the extended Nijboer-Zernike theory,” J. Eur. Opt. Soc. Rapid Publ. 4, 09048 (2009).
[CrossRef]

2008

R. Aarts, J. J. M. Braat, P. Dirksen, S. van Haver, C. van Heesch, and A. Janssen, “Analytic expressions and approximations for the on-axis, aberration-free Rayleigh and Debye integral in the case of focusing fields on a circular aperture,” J. Eur. Opt. Soc. Rapid Publ. 3, 08039 (2008).
[CrossRef]

H. P. Urbach and S. F. Pereira, “The field in focus with maximum longitudinal electric component,” Phys. Rev. Lett. 100, 1233904 (2008).
[CrossRef]

2007

A. J. E. M. Janssen, S. van Haver, J. J. M. Braat, and P. Dirksen, “Strehl ratio and optimum focus of high-numerical-aperture beams,” J. Eur. Opt. Soc. Rapid Publ. 2, 07008 (2007).
[CrossRef]

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A: Mater. Sci. Process. 86, 329–334 (2007).
[CrossRef]

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

2005

2004

2003

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

2002

2001

L. Novotny, M. R. Beverluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light–theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

2000

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

1997

V. Dhayalan and J. J. Stamnes, “Focusing of mixed-dipole waves,” Pure Appl. Opt. 6, 317–345 (1997).
[CrossRef]

C. J. R. Sheppard and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik 104, 175–177 (1997).

1996

1994

C. J. R. Sheppard and K. G. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41, 1495–1505 (1994).
[CrossRef]

X. S. Xie and R. C. Dunn, “Probing single molecule dynamics,” Science 265, 361–364 (1994).
[CrossRef] [PubMed]

1981

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

1959

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral repesentation of the image field,” Proc. R. Soc. London, Ser. A 253, 349–357 (1959).
[CrossRef]

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. A 253, 358–379 (1959).
[CrossRef]

1920

V. S. Ignatowsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans. Opt. Inst. Petrograd , I, paper V (1920), (in Russian).

1919

V. S. Ignatowsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd I, paper IV (1919), (in Russian).

Aarts, R.

R. Aarts, J. J. M. Braat, P. Dirksen, S. van Haver, C. van Heesch, and A. Janssen, “Analytic expressions and approximations for the on-axis, aberration-free Rayleigh and Debye integral in the case of focusing fields on a circular aperture,” J. Eur. Opt. Soc. Rapid Publ. 3, 08039 (2008).
[CrossRef]

Audouard, E.

Beverluis, M. R.

L. Novotny, M. R. Beverluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

Braat, J. J. M.

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and S. F. Pereira, “Image formation in a multilayer using the extended Nijboer-Zernike theory,” J. Eur. Opt. Soc. Rapid Publ. 4, 09048 (2009).
[CrossRef]

R. Aarts, J. J. M. Braat, P. Dirksen, S. van Haver, C. van Heesch, and A. Janssen, “Analytic expressions and approximations for the on-axis, aberration-free Rayleigh and Debye integral in the case of focusing fields on a circular aperture,” J. Eur. Opt. Soc. Rapid Publ. 3, 08039 (2008).
[CrossRef]

A. J. E. M. Janssen, S. van Haver, J. J. M. Braat, and P. Dirksen, “Strehl ratio and optimum focus of high-numerical-aperture beams,” J. Eur. Opt. Soc. Rapid Publ. 2, 07008 (2007).
[CrossRef]

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in “Progress in Optics ,” Vol. 51, E. Wolf, ed. (Elsevier B.V., 2008), chap. 6, pp. 349–468.
[CrossRef]

Brown, T. G.

L. Novotny, M. R. Beverluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

Chen, W.

Choudhurry, A.

Dhayalan, V.

V. Dhayalan and J. J. Stamnes, “Focusing of mixed-dipole waves,” Pure Appl. Opt. 6, 317–345 (1997).
[CrossRef]

Dirksen, P.

R. Aarts, J. J. M. Braat, P. Dirksen, S. van Haver, C. van Heesch, and A. Janssen, “Analytic expressions and approximations for the on-axis, aberration-free Rayleigh and Debye integral in the case of focusing fields on a circular aperture,” J. Eur. Opt. Soc. Rapid Publ. 3, 08039 (2008).
[CrossRef]

A. J. E. M. Janssen, S. van Haver, J. J. M. Braat, and P. Dirksen, “Strehl ratio and optimum focus of high-numerical-aperture beams,” J. Eur. Opt. Soc. Rapid Publ. 2, 07008 (2007).
[CrossRef]

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in “Progress in Optics ,” Vol. 51, E. Wolf, ed. (Elsevier B.V., 2008), chap. 6, pp. 349–468.
[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]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light–theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

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

Dunn, R. C.

X. S. Xie and R. C. Dunn, “Probing single molecule dynamics,” Science 265, 361–364 (1994).
[CrossRef] [PubMed]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light–theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

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

Feurer, T.

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A: Mater. Sci. Process. 86, 329–334 (2007).
[CrossRef]

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light–theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

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

Hecht, B.

L. Novotny and B. Hecht, “Principles of nano-optics,” Section 2.10.2, (Cambridge University Press, 2008).

Helseth, L. E.

L. E. Helseth, “Focusing of atoms with strongly confined light potentials,” Opt. Commun. 212, 343–352 (2002).
[CrossRef]

Huignard, J.-P.

Huot, N.

Iglesias, I.

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

Ignatowsky, V. S.

V. S. Ignatowsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans. Opt. Inst. Petrograd , I, paper V (1920), (in Russian).

V. S. Ignatowsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd I, paper IV (1919), (in Russian).

Janssen, A.

R. Aarts, J. J. M. Braat, P. Dirksen, S. van Haver, C. van Heesch, and A. Janssen, “Analytic expressions and approximations for the on-axis, aberration-free Rayleigh and Debye integral in the case of focusing fields on a circular aperture,” J. Eur. Opt. Soc. Rapid Publ. 3, 08039 (2008).
[CrossRef]

Janssen, A. J. E. M.

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and S. F. Pereira, “Image formation in a multilayer using the extended Nijboer-Zernike theory,” J. Eur. Opt. Soc. Rapid Publ. 4, 09048 (2009).
[CrossRef]

A. J. E. M. Janssen, S. van Haver, J. J. M. Braat, and P. Dirksen, “Strehl ratio and optimum focus of high-numerical-aperture beams,” J. Eur. Opt. Soc. Rapid Publ. 2, 07008 (2007).
[CrossRef]

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in “Progress in Optics ,” Vol. 51, E. Wolf, ed. (Elsevier B.V., 2008), chap. 6, pp. 349–468.
[CrossRef]

Juskaitis, R.

Larat, C.

Larkin, K. G.

C. J. R. Sheppard and K. G. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41, 1495–1505 (1994).
[CrossRef]

Leger, J.

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]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light–theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

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

Li, Y.

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

Loiseaux, B.

Massoumian, F.

Meier, M.

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A: Mater. Sci. Process. 86, 329–334 (2007).
[CrossRef]

Neil, M. A. A.

Novotny, L.

L. Novotny, M. R. Beverluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

L. Novotny and B. Hecht, “Principles of nano-optics,” Section 2.10.2, (Cambridge University Press, 2008).

Pereira, S. F.

H. P. Urbach and S. F. Pereira, “Erratum: focused fields of given power with maximum electric field components,” Phys. Rev. A 81, 059903 (2010).
[CrossRef]

H. P. Urbach and S. F. Pereira, “Focused fields of given power with maximum electric field components,” Phys. Rev. A 79, 013825 (2009).
[CrossRef]

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and S. F. Pereira, “Image formation in a multilayer using the extended Nijboer-Zernike theory,” J. Eur. Opt. Soc. Rapid Publ. 4, 09048 (2009).
[CrossRef]

H. P. Urbach and S. F. Pereira, “The field in focus with maximum longitudinal electric component,” Phys. Rev. Lett. 100, 1233904 (2008).
[CrossRef]

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]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light–theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 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. A 253, 358–379 (1959).
[CrossRef]

Romano, V.

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A: Mater. Sci. Process. 86, 329–334 (2007).
[CrossRef]

Sanner, N.

Schadt, M.

Sheppard, C. J. R.

C. J. R. Sheppard and A. Choudhurry, “Annular pupils, radial polarization and superresolution,” Appl. Opt. 43, 4322–4327 (2004).
[CrossRef] [PubMed]

C. J. R. Sheppard and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik 104, 175–177 (1997).

C. J. R. Sheppard and K. G. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41, 1495–1505 (1994).
[CrossRef]

Stalder, M.

Stamnes, J. J.

V. Dhayalan and J. J. Stamnes, “Focusing of mixed-dipole waves,” Pure Appl. Opt. 6, 317–345 (1997).
[CrossRef]

Török, P.

C. J. R. Sheppard and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik 104, 175–177 (1997).

Urbach, H. P.

H. P. Urbach and S. F. Pereira, “Erratum: focused fields of given power with maximum electric field components,” Phys. Rev. A 81, 059903 (2010).
[CrossRef]

H. P. Urbach and S. F. Pereira, “Focused fields of given power with maximum electric field components,” Phys. Rev. A 79, 013825 (2009).
[CrossRef]

H. P. Urbach and S. F. Pereira, “The field in focus with maximum longitudinal electric component,” Phys. Rev. Lett. 100, 1233904 (2008).
[CrossRef]

van Haver, S.

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and S. F. Pereira, “Image formation in a multilayer using the extended Nijboer-Zernike theory,” J. Eur. Opt. Soc. Rapid Publ. 4, 09048 (2009).
[CrossRef]

R. Aarts, J. J. M. Braat, P. Dirksen, S. van Haver, C. van Heesch, and A. Janssen, “Analytic expressions and approximations for the on-axis, aberration-free Rayleigh and Debye integral in the case of focusing fields on a circular aperture,” J. Eur. Opt. Soc. Rapid Publ. 3, 08039 (2008).
[CrossRef]

A. J. E. M. Janssen, S. van Haver, J. J. M. Braat, and P. Dirksen, “Strehl ratio and optimum focus of high-numerical-aperture beams,” J. Eur. Opt. Soc. Rapid Publ. 2, 07008 (2007).
[CrossRef]

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in “Progress in Optics ,” Vol. 51, E. Wolf, ed. (Elsevier B.V., 2008), chap. 6, pp. 349–468.
[CrossRef]

van Heesch, C.

R. Aarts, J. J. M. Braat, P. Dirksen, S. van Haver, C. van Heesch, and A. Janssen, “Analytic expressions and approximations for the on-axis, aberration-free Rayleigh and Debye integral in the case of focusing fields on a circular aperture,” J. Eur. Opt. Soc. Rapid Publ. 3, 08039 (2008).
[CrossRef]

Vohnsen, B.

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

Wang, J.

Wilson, T.

Wolf, E.

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

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. A 253, 358–379 (1959).
[CrossRef]

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral repesentation of the image field,” Proc. R. Soc. London, Ser. A 253, 349–357 (1959).
[CrossRef]

Xie, X. S.

X. S. Xie and R. C. Dunn, “Probing single molecule dynamics,” Science 265, 361–364 (1994).
[CrossRef] [PubMed]

Youngworth, K. S.

L. Novotny, M. R. Beverluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

Zhan, Q.

Zhan, Q. W.

Appl. Opt.

Appl. Phys. A: Mater. Sci. Process.

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A: Mater. Sci. Process. 86, 329–334 (2007).
[CrossRef]

Appl. Phys. B

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light–theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

J. Eur. Opt. Soc. Rapid Publ.

R. Aarts, J. J. M. Braat, P. Dirksen, S. van Haver, C. van Heesch, and A. Janssen, “Analytic expressions and approximations for the on-axis, aberration-free Rayleigh and Debye integral in the case of focusing fields on a circular aperture,” J. Eur. Opt. Soc. Rapid Publ. 3, 08039 (2008).
[CrossRef]

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and S. F. Pereira, “Image formation in a multilayer using the extended Nijboer-Zernike theory,” J. Eur. Opt. Soc. Rapid Publ. 4, 09048 (2009).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Spherical coordinates in image space with respect to wave unit vector .

Fig. 2
Fig. 2

The relationship between the optimization angle ϑ v for which problem 1 is solved and angle ϑ u between the direction of the optimum electric field vector E(0) and the z-axis, for several values of the numerical aperture NA.

Fig. 3
Fig. 3

Left: The electric field E 1 v ^ (0) projected along and û as functions of ϑ v for several values of the numerical aperture. : The projections of the optimum electric fields E 1 v ^ (0) and E 2 v ^ (0) along the direction , as function of ϑ v for several values of the numerical aperture.

Fig. 4
Fig. 4

The focusing lens with numerical aperture NA.

Fig. 5
Fig. 5

Left: Snapshots of the electric field in the entrance pupil, E p , which after focusing yields the field with maximum component along obtained by solving 1() (left) and the optimum pupil field that is solution of 2() (right), where is such that vy = 0 and ϑ v = 45°. In both cases NA/n = 0.75.

Fig. 6
Fig. 6

The two lens system that is used to image the optimum electric field in image space when a dipole source is placed in object space. The focal length f 1 denotes the radius of entrance pupil sphere 1, focal length f 2 that of exit pupil sphere 2.

Fig. 7
Fig. 7

Schematics of the imaging of a dipole using the two lens system with (paraxial) magnification equal to 1. The dipole produces the electric field at the focal point which has maximum amplitude in the direction of and which has maximum amplitude and is linearly polarized in the direction of û.

Equations (61)

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( r , t ) = Re [ E ( r ) e i ω t ] = 1 4 π 2 Re [ k x 2 + k y 2 k 0 N A A ( k x , k y ) e i ( k · r ω t ) d k x d k y ] ,
A ( k x , k y ) · k = 0.
P ( A ) = 1 2 Re [ E ( r ) × H ( r ) * ] · z ^ d x d y = 1 8 π 2 k 0 ( ɛ 0 μ 0 ) 1 / 2 Ω | A ( k x , k y ) | 2 k z d k x d k y .
( 0 , t ) · v ^ = Re [ E ( 0 ) · v ^ e i ω t ] .
Im [ E ( 0 ) · v ^ ] = 1 4 π 2 Ω Im [ A ( k x , k y ) · v ^ ] d k x d k y = 0.
1 ( v ^ ) : max A 0 E v ( A ) E ( 0 ) · v ^ , under the constraint P ( A ) P 0 ,
A ( k x , k y ) = ( 8 π P 0 n ) 1 / 2 ( μ 0 ɛ 0 ) 1 / 4 1 k z 1 Γ ( α max , v z ) [ v ^ ( v ^ · k ^ ) k ^ ] ,
Γ ( α max , v z ) = { 4 3 cos α max 1 3 cos 3 α max v z 2 cos α max sin 2 α max } 1 / 2 ,
sin α max = NA / n .
k x = k sin α cos β ,
k y = k sin α sin β ,
k z = k cos α ,
E ( ρ , φ , z ) = 2 π n P 0 λ 0 ( μ 0 ɛ 0 ) 1 / 4 1 Γ ( α max , v z ) ¯ ¯ ( ρ , φ , z ) ( v x v y v z ) ,
¯ ¯ ( ρ , φ , z ) = ( g 0 0 , 1 ( ρ , z ) + g 0 2 , 1 ( ρ , z ) 0 0 0 g 0 0 , 1 ( ρ , z ) + g 0 2 , 1 ( ρ , z ) 0 0 0 2 g 0 0 , 3 ( ρ , z ) ) + ( g 2 0 , 3 ( ρ , z ) cos ( 2 φ ) g 2 0 , 3 ( ρ , z ) sin ( 2 φ ) 2 i g 1 1 , 2 ( ρ , z ) cos φ g 2 0 , 3 ( ρ , z ) sin ( 2 φ ) g 2 0 , 3 ( ρ , z ) cos ( 2 φ ) 2 i g 1 1 , 2 ( ρ , z ) sin φ 2 i g 1 1 , 2 ( ρ , z ) cos φ 2 i g 1 1 , 2 ( ρ , z ) sin φ 0 ) ,
g l ν , μ ( ρ , z ) = 0 α max e i k 0 n z cos α cos ν α sin μ α J l ( k 0 n ρ sin α ) d α .
E ( 0 ) = [ g 0 0 , 1 ( 0 ) + g 0 2 , 1 ( 0 ) ] v x x ^ + [ g 0 0 , 1 ( 0 ) + g 0 2 , 1 ( 0 ) ] v y y ^ + 2 g 0 0 , 3 ( ρ , z ) v z z ^ ,
g 0 0 , 1 ( 0 ) = 0 α max sin α d α = 1 cos α max ,
g 0 2 , 1 ( 0 ) = 0 α max cos 2 α sin α d α = 1 3 1 3 cos 3 α max ,
g 0 0 , 3 ( 0 ) = 0 α max sin 3 α d α = 2 3 cos α max + 1 3 cos 3 α max .
γ + g 0 0 , 1 ( 0 ) + g 0 2 , 1 ( 0 ) = 4 3 cos α max 1 3 cos 3 α max ,
γ 2 [ g 0 0 , 1 ( 0 ) g 0 2 , 1 ( 0 ) ] = 4 3 2 cos α max + 2 3 cos 3 α max .
Γ ( α max , v z ) = { γ + ( γ + γ ) v z 2 } 1 / 2 ,
E ( 0 ) = γ + ( v x x ^ + v y y ^ ) + γ v z z ^ .
u ^ = E ( 0 ) | E ( 0 ) | = γ + ( v x x ^ + v y y ^ ) + γ v z z ^ γ + 2 ( v x 2 + v y 2 ) + γ 2 v z 2 .
v x = sin ϑ v , v z = cos ϑ v ,
u x = sin ϑ u , u z = cos ϑ u ,
E ( 0 ) · ( v ^ × z ^ ) = 0 and E ( 0 ) · [ v ^ × ( v ^ × z ^ ) ] = 0.
2 ( v ^ ) : max A 0 E v ( A ) = E ( 0 ) · v ^ , under the constraints P ( A ) P 0 and ( 26 ) ,
E 1 v ^ = E 2 u ^ .
0 < E 1 v ^ ( 0 ) · u ^ E 2 u ^ ( 0 ) · u ^ .
E 1 v ^ ( 0 ) · v ^ E 2 u ^ ( 0 ) · v ^ .
v ^ = γ ( u x x ^ + u y y ^ ) + γ + u z z ^ γ 2 ( u x 2 + u y 2 ) + γ + 2 u z 2 = γ u ^ + ( γ + γ ) u z z ^ γ 2 ( u x 2 + u y 2 ) + γ + 2 u z 2 .
A ( k x , k y ) = ( 8 π P 0 n ) 1 / 2 ( μ 0 ɛ 0 ) 1 / 4 1 k z 1 Γ ( α max , v z ) [ v ^ ( v ^ · k ^ ) k ^ ] = ( 8 π P 0 n ) 1 / 2 ( μ 0 ɛ 0 ) 1 / 4 1 k z 1 Γ ˜ ( α max , u z ) × { γ [ u ^ ( u ^ · k ^ ) k ^ ] + ( γ + γ ) u z [ z ^ ( z ^ · k ^ ) k ^ ] } ,
Γ ˜ ( α max , u z ) Γ ( α max , v z ) γ 2 ( u x 2 + u y 2 ) + γ + 2 u z 2 = γ + γ { γ ( γ + γ ) u z 2 } 1 / 2 ,
E ( ρ , φ , z ) = 2 π n P 0 λ 0 ( μ 0 ɛ 0 ) 1 / 4 1 Γ ˜ ( α max , u z ) ¯ ¯ ( ρ , φ , z ) ( γ u x γ u y γ + u z ) ,
x p = ρ p cos φ p , y p = ρ p sin φ p .
ρ ^ p = cos φ p x ^ p + sin φ p y ^ p ,
φ ^ p = sin φ p x ^ p + cos φ p · y ^ p ,
E p ( ρ p , φ p ) = E ρ p ( ρ p , φ p ) ρ ^ p + E φ p ( ρ p , φ p ) φ ^ p ,
f λ 0 n sin 2 ( α max ) ,
k x = k x p f = k ρ p f cos φ p , k y = k y p f = k ρ p f sin φ p .
i S ^ ( k ^ ) = k ^ × z ^ | k ^ × z ^ | = 1 k x 2 + k y 2 ( k y k x 0 ) ,
i P ^ ( k ^ ) = i S ^ ( k ^ ) × k ^ | i S ^ ( k ^ ) × k ^ | = 1 k 1 k x 2 + k y 2 ( k x k z k y k z k x 2 + k y 2 ) .
A ( k x , k y ) = A P ( k x , k y ) i P ^ ( k ^ ) + A S ( k x , k y ) i S ^ ( k ^ ) .
E ρ p ( ρ p , φ p ) = k k z 2 π i f A P ( k ρ p f cos φ p , k ρ p f sin φ p ) ,
E φ p ( ρ p , φ p ) = k k z 2 π i f A S ( k ρ p f cos φ p , k ρ p f sin φ p ) ,
k z = k 1 ρ p 2 f 2 .
E p ( ρ p , φ p ) = i ( 2 P 0 π n ) 1 / 2 ( μ 0 ɛ 0 ) 1 / 4 1 f Γ ( α max , v z ) × { [ ( f 2 ρ p 2 ) 1 / 4 f 1 / 2 ( cos φ p v x + sin φ p v y ) + ρ p f 1 / 2 ( f 2 ρ p 2 ) 1 / 4 v z ] ρ ^ p + f 1 / 2 ( f 2 ρ p 2 ) 1 / 4 ( cos φ p v y sin φ p v x ) φ ^ p } .
E p ( ρ p , φ p ) = i ( 2 P 0 π n ) 1 / 2 ( μ 0 ɛ 0 ) 1 / 4 1 f Γ ˜ ( α max , u z ) × { [ ( f 2 ρ p 2 ) 1 / 4 f 1 / 2 γ ( cos φ p u x + sin φ p u y ) + ρ p f 1 / 2 ( f 2 ρ p 2 ) 1 / 4 γ + u z ] ρ ^ p + f 1 / 2 ( f 2 ρ p 2 ) 1 / 4 γ ( cos φ p u y sin φ p u x ) φ ^ p } .
E d ( r ) = k 2 ɛ 0 p G ( r ) ( · p ɛ 0 G ( r ) ) ,
G ( r ) = e i k r 4 π r .
[ E d ] ( k x o , k y o , z ) = [ k 2 p ( k o · p ) k o ] e i k z o z 2 i ɛ 0 k z o ,
A o ( k x o , k y o ) = [ E d ] ( k x o , k y o , 0 )
= 1 2 i ɛ 0 k z o [ k 2 p ( k o · p ) k o ] .
A i ( k x i , k y i ) · k ^ i = A o ( k x o , k y o ) · k ^ o = 0 ,
A i ( k x i , k y i ) · i P ^ i = A o ( k x o , k y o ) · i P ^ o ,
A i ( k x i , k y i ) · i S ^ i = A o ( k x o , k y o ) · i S ^ o ,
A i ( k x , k y ) = 1 2 i ɛ 0 k z [ k 2 p ˜ ( p ˜ · k ) k ] ,
| p ˜ | = ( 32 π ɛ 0 P 0 n ) 1 / 2 ( ɛ 0 μ 0 ) 1 / 4 Γ ( α max , v z ) ,
v ^ = ( v x , v y , v z ) = ( p x , p y , p z ) / | p ˜ | = p ˜ / | p ˜ | ,
u ^ = γ + ( p x x ^ + p y y ^ ) γ p z z ^ γ + 2 ( p x 2 + p y 2 ) + γ 2 p z 2 ,

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