Abstract

Rigorous vectorial focusing theory is used to study the imaging of small adjacent particles with a confocal laser scanning system. We consider radially polarized illumination with an optimized amplitude distribution and an annular lens to obtain a narrower distribution of the longitudinal component of the field in focus. A polarization convertor at the detector side is added to transform radial polarization to linear polarization in order to make the signal detectable with a single mode fiber.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. 19, 780–782 (1994).
    [Crossref] [PubMed]
  2. E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
    [Crossref] [PubMed]
  3. M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (storm),” Nat. Methods 3, 793 (2006).
    [Crossref] [PubMed]
  4. D. V. Patel and C. N. McGhee, “Contemporary in vivo confocal microscopy of the living human cornea using white light and laser scanning techniques: a major review,” Clin. Exp. Ophthalmol. 35, 71–88 (2007).
    [Crossref] [PubMed]
  5. A. Hoffman, M. Goetz, M. Vieth, P. Galle, M. Neurath, and R. Kiesslich, “Confocal laser endomicroscopy: technical status and current indications,” Endoscopy 38, 1275–1283 (2006).
    [Crossref] [PubMed]
  6. S. Le Person, J. Puiggali, M. Baron, and M. Roques, “Near infrared drying of pharmaceutical thin films: experimental analysis of internal mass transport,” Chem. Eng. Process. Process. Intensif. 37, 257–263 (1998).
    [Crossref]
  7. S. Götzinger, L. de S Menezes, O. Benson, D. Talapin, N. Gaponik, H. Weller, A. Rogach, and V. Sandoghdar, “Confocal microscopy and spectroscopy of nanocrystals on a high-q microsphere resonator,” J. Opt. B 6, 154 (2004).
    [Crossref]
  8. K. Lindfors, T. Kalkbrenner, P. Stoller, and V. Sandoghdar, “Detection and spectroscopy of gold nanoparticles using supercontinuum white light confocal microscopy,” Phys. Rev. Lett. 93, 037401 (2004).
    [Crossref] [PubMed]
  9. L. Novotny and B. Hecht, Principles of nano-optics (Cambridge University).
  10. T. Wilson, R. Juškaitis, and P. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes,” Opt. Commun. 141, 298 – 313 (1997).
    [Crossref]
  11. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
    [Crossref] [PubMed]
  12. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000).
    [Crossref] [PubMed]
  13. R. Chen, K. Agarwal, C. J. Sheppard, and X. Chen, “Imaging using cylindrical vector beams in a high-numerical-aperture microscopy system,” Opt. Lett. 38, 3111–3114 (2013).
    [Crossref] [PubMed]
  14. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1 – 7 (2000).
    [Crossref]
  15. C. J. Sheppard and A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Appl. Opt. 43, 4322–4327 (2004).
    [Crossref] [PubMed]
  16. L. Yang, X. Xie, S. Wang, and J. Zhou, “Minimized spot of annular radially polarized focusing beam,” Opt. Lett. 38, 1331–1333 (2013).
    [Crossref] [PubMed]
  17. Y. Kozawa and S. Sato, “Focusing property of a double-ring-shaped radially polarized beam,” Opt. Lett. 31, 820–822 (2006).
    [Crossref] [PubMed]
  18. 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 (2008).
    [Crossref]
  19. H. Guo, X. Weng, M. Jiang, Y. Zhao, G. Sui, Q. Hu, Y. Wang, and S. Zhuang, “Tight focusing of a higher-order radially polarized beam transmitting through multi-zone binary phase pupil filters,” Opt. Express 21, 5363–5372 (2013).
    [Crossref] [PubMed]
  20. F. Tang, Y. Wang, L. Qiu, W. Zhao, and Y. Sun, “Super-resolution radially polarized-light pupil-filtering confocal sensing technology,” Appl. Opt. 53, 7407–7414 (2014).
    [Crossref] [PubMed]
  21. N. Davidson and N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29, 1318–1320 (2004).
    [Crossref] [PubMed]
  22. H. Urbach and S. Pereira, “Field in focus with a maximum longitudinal electric component,” Phys. Rev. Lett. 100, 123904 (2008).
    [Crossref] [PubMed]
  23. K. Ushakova, Q. Van den Berg, S. Pereira, and H. Urbach, “Demonstration of spot size reduction by focussing amplitude modulated radially polarized light on a photoresist,” J. Opt. 17, 125615 (2015).
    [Crossref]
  24. D. P. Biss, K. S. Youngworth, and T. G. Brown, “Dark-field imaging with cylindrical-vector beams,” Appl. Opt. 45, 470–479 (2006).
    [Crossref] [PubMed]
  25. W. T. Tang, E. Y. Yew, and C. J. Sheppard, “Polarization conversion in confocal microscopy with radially polarized illumination,” Opt. Lett. 34, 2147–2149 (2009).
    [Crossref] [PubMed]
  26. X. Xie, Y. Chen, K. Yang, and J. Zhou, “Harnessing the point-spread function for high-resolution far-field optical microscopy,” Phys. Rev. Lett. 113, 263901 (2014).
    [Crossref]
  27. V. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” Trans. Opt. Inst 1, 1–36 (1919).
  28. E. Wolf, “Electromagnetic diffraction in optical systems. i. an integral representation of the image field,” in Proc. Royal Soc. London A: Math. Phys. Eng. Sci., vol. 253 (The Royal Society, 1959), pp. 349–357.
    [Crossref]
  29. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. ii. structure of the image field in an aplanatic system,” in Proc. Royal Soc. London A: Math. Phys. Eng. Sci., vol. 253 (The Royal Society, 1959), pp. 358–379.
    [Crossref]
  30. M. Abramowitz and I. A. Stegun, Handbook of mathematical functions: with formulas, graphs, and mathematical tables, vol. 55 (Courier Corporation).
  31. J. Jackson, Classical Electrodynamics, 2nd ed.(John Wiley & Sons).
  32. P. Török, P. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
    [Crossref]
  33. D. B. Murphy, Fundamentals of Light Microscopy and Electronic Imaging (John Wiley & Sons).

2015 (1)

K. Ushakova, Q. Van den Berg, S. Pereira, and H. Urbach, “Demonstration of spot size reduction by focussing amplitude modulated radially polarized light on a photoresist,” J. Opt. 17, 125615 (2015).
[Crossref]

2014 (2)

X. Xie, Y. Chen, K. Yang, and J. Zhou, “Harnessing the point-spread function for high-resolution far-field optical microscopy,” Phys. Rev. Lett. 113, 263901 (2014).
[Crossref]

F. Tang, Y. Wang, L. Qiu, W. Zhao, and Y. Sun, “Super-resolution radially polarized-light pupil-filtering confocal sensing technology,” Appl. Opt. 53, 7407–7414 (2014).
[Crossref] [PubMed]

2013 (3)

2009 (1)

2008 (2)

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 (2008).
[Crossref]

H. Urbach and S. Pereira, “Field in focus with a maximum longitudinal electric component,” Phys. Rev. Lett. 100, 123904 (2008).
[Crossref] [PubMed]

2007 (1)

D. V. Patel and C. N. McGhee, “Contemporary in vivo confocal microscopy of the living human cornea using white light and laser scanning techniques: a major review,” Clin. Exp. Ophthalmol. 35, 71–88 (2007).
[Crossref] [PubMed]

2006 (5)

A. Hoffman, M. Goetz, M. Vieth, P. Galle, M. Neurath, and R. Kiesslich, “Confocal laser endomicroscopy: technical status and current indications,” Endoscopy 38, 1275–1283 (2006).
[Crossref] [PubMed]

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref] [PubMed]

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (storm),” Nat. Methods 3, 793 (2006).
[Crossref] [PubMed]

D. P. Biss, K. S. Youngworth, and T. G. Brown, “Dark-field imaging with cylindrical-vector beams,” Appl. Opt. 45, 470–479 (2006).
[Crossref] [PubMed]

Y. Kozawa and S. Sato, “Focusing property of a double-ring-shaped radially polarized beam,” Opt. Lett. 31, 820–822 (2006).
[Crossref] [PubMed]

2004 (4)

N. Davidson and N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29, 1318–1320 (2004).
[Crossref] [PubMed]

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

S. Götzinger, L. de S Menezes, O. Benson, D. Talapin, N. Gaponik, H. Weller, A. Rogach, and V. Sandoghdar, “Confocal microscopy and spectroscopy of nanocrystals on a high-q microsphere resonator,” J. Opt. B 6, 154 (2004).
[Crossref]

K. Lindfors, T. Kalkbrenner, P. Stoller, and V. Sandoghdar, “Detection and spectroscopy of gold nanoparticles using supercontinuum white light confocal microscopy,” Phys. Rev. Lett. 93, 037401 (2004).
[Crossref] [PubMed]

2003 (1)

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

2000 (2)

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

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

1998 (2)

P. Török, P. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[Crossref]

S. Le Person, J. Puiggali, M. Baron, and M. Roques, “Near infrared drying of pharmaceutical thin films: experimental analysis of internal mass transport,” Chem. Eng. Process. Process. Intensif. 37, 257–263 (1998).
[Crossref]

1997 (1)

T. Wilson, R. Juškaitis, and P. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes,” Opt. Commun. 141, 298 – 313 (1997).
[Crossref]

1994 (1)

1919 (1)

V. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” Trans. Opt. Inst 1, 1–36 (1919).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of mathematical functions: with formulas, graphs, and mathematical tables, vol. 55 (Courier Corporation).

Agarwal, K.

Baron, M.

S. Le Person, J. Puiggali, M. Baron, and M. Roques, “Near infrared drying of pharmaceutical thin films: experimental analysis of internal mass transport,” Chem. Eng. Process. Process. Intensif. 37, 257–263 (1998).
[Crossref]

Bates, M.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (storm),” Nat. Methods 3, 793 (2006).
[Crossref] [PubMed]

Benson, O.

S. Götzinger, L. de S Menezes, O. Benson, D. Talapin, N. Gaponik, H. Weller, A. Rogach, and V. Sandoghdar, “Confocal microscopy and spectroscopy of nanocrystals on a high-q microsphere resonator,” J. Opt. B 6, 154 (2004).
[Crossref]

Betzig, E.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref] [PubMed]

Biss, D. P.

Bokor, N.

Bonifacino, J. S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref] [PubMed]

Brown, T. G.

Chen, R.

Chen, X.

Chen, Y.

X. Xie, Y. Chen, K. Yang, and J. Zhou, “Harnessing the point-spread function for high-resolution far-field optical microscopy,” Phys. Rev. Lett. 113, 263901 (2014).
[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 (2008).
[Crossref]

Choudhury, A.

Davidson, M. W.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref] [PubMed]

Davidson, N.

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, “Focusing light to a tighter spot,” Opt. Commun. 179, 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 – 7 (2000).
[Crossref]

Galle, P.

A. Hoffman, M. Goetz, M. Vieth, P. Galle, M. Neurath, and R. Kiesslich, “Confocal laser endomicroscopy: technical status and current indications,” Endoscopy 38, 1275–1283 (2006).
[Crossref] [PubMed]

Gaponik, N.

S. Götzinger, L. de S Menezes, O. Benson, D. Talapin, N. Gaponik, H. Weller, A. Rogach, and V. Sandoghdar, “Confocal microscopy and spectroscopy of nanocrystals on a high-q microsphere resonator,” J. Opt. B 6, 154 (2004).
[Crossref]

Glöckl, O.

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

Goetz, M.

A. Hoffman, M. Goetz, M. Vieth, P. Galle, M. Neurath, and R. Kiesslich, “Confocal laser endomicroscopy: technical status and current indications,” Endoscopy 38, 1275–1283 (2006).
[Crossref] [PubMed]

Götzinger, S.

S. Götzinger, L. de S Menezes, O. Benson, D. Talapin, N. Gaponik, H. Weller, A. Rogach, and V. Sandoghdar, “Confocal microscopy and spectroscopy of nanocrystals on a high-q microsphere resonator,” J. Opt. B 6, 154 (2004).
[Crossref]

Guo, H.

Hecht, B.

L. Novotny and B. Hecht, Principles of nano-optics (Cambridge University).

Hell, S. W.

Hess, H. F.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref] [PubMed]

Higdon, P.

P. Török, P. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[Crossref]

T. Wilson, R. Juškaitis, and P. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes,” Opt. Commun. 141, 298 – 313 (1997).
[Crossref]

Hoffman, A.

A. Hoffman, M. Goetz, M. Vieth, P. Galle, M. Neurath, and R. Kiesslich, “Confocal laser endomicroscopy: technical status and current indications,” Endoscopy 38, 1275–1283 (2006).
[Crossref] [PubMed]

Hu, Q.

Ignatowsky, V.

V. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” Trans. Opt. Inst 1, 1–36 (1919).

Jackson, J.

J. Jackson, Classical Electrodynamics, 2nd ed.(John Wiley & Sons).

Jiang, M.

Juškaitis, R.

T. Wilson, R. Juškaitis, and P. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes,” Opt. Commun. 141, 298 – 313 (1997).
[Crossref]

Kalkbrenner, T.

K. Lindfors, T. Kalkbrenner, P. Stoller, and V. Sandoghdar, “Detection and spectroscopy of gold nanoparticles using supercontinuum white light confocal microscopy,” Phys. Rev. Lett. 93, 037401 (2004).
[Crossref] [PubMed]

Kiesslich, R.

A. Hoffman, M. Goetz, M. Vieth, P. Galle, M. Neurath, and R. Kiesslich, “Confocal laser endomicroscopy: technical status and current indications,” Endoscopy 38, 1275–1283 (2006).
[Crossref] [PubMed]

Kozawa, Y.

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, “Focusing light to a tighter spot,” Opt. Commun. 179, 1 – 7 (2000).
[Crossref]

Lindfors, K.

K. Lindfors, T. Kalkbrenner, P. Stoller, and V. Sandoghdar, “Detection and spectroscopy of gold nanoparticles using supercontinuum white light confocal microscopy,” Phys. Rev. Lett. 93, 037401 (2004).
[Crossref] [PubMed]

Lindwasser, O. W.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref] [PubMed]

Lippincott-Schwartz, J.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[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, 501 (2008).
[Crossref]

McGhee, C. N.

D. V. Patel and C. N. McGhee, “Contemporary in vivo confocal microscopy of the living human cornea using white light and laser scanning techniques: a major review,” Clin. Exp. Ophthalmol. 35, 71–88 (2007).
[Crossref] [PubMed]

Menezes, L. de S

S. Götzinger, L. de S Menezes, O. Benson, D. Talapin, N. Gaponik, H. Weller, A. Rogach, and V. Sandoghdar, “Confocal microscopy and spectroscopy of nanocrystals on a high-q microsphere resonator,” J. Opt. B 6, 154 (2004).
[Crossref]

Murphy, D. B.

D. B. Murphy, Fundamentals of Light Microscopy and Electronic Imaging (John Wiley & Sons).

Neurath, M.

A. Hoffman, M. Goetz, M. Vieth, P. Galle, M. Neurath, and R. Kiesslich, “Confocal laser endomicroscopy: technical status and current indications,” Endoscopy 38, 1275–1283 (2006).
[Crossref] [PubMed]

Novotny, L.

L. Novotny and B. Hecht, Principles of nano-optics (Cambridge University).

Olenych, S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref] [PubMed]

Patel, D. V.

D. V. Patel and C. N. McGhee, “Contemporary in vivo confocal microscopy of the living human cornea using white light and laser scanning techniques: a major review,” Clin. Exp. Ophthalmol. 35, 71–88 (2007).
[Crossref] [PubMed]

Patterson, G. H.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref] [PubMed]

Pereira, S.

K. Ushakova, Q. Van den Berg, S. Pereira, and H. Urbach, “Demonstration of spot size reduction by focussing amplitude modulated radially polarized light on a photoresist,” J. Opt. 17, 125615 (2015).
[Crossref]

H. Urbach and S. Pereira, “Field in focus with a maximum longitudinal electric component,” Phys. Rev. Lett. 100, 123904 (2008).
[Crossref] [PubMed]

Person, S. Le

S. Le Person, J. Puiggali, M. Baron, and M. Roques, “Near infrared drying of pharmaceutical thin films: experimental analysis of internal mass transport,” Chem. Eng. Process. Process. Intensif. 37, 257–263 (1998).
[Crossref]

Puiggali, J.

S. Le Person, J. Puiggali, M. Baron, and M. Roques, “Near infrared drying of pharmaceutical thin films: experimental analysis of internal mass transport,” Chem. Eng. Process. Process. Intensif. 37, 257–263 (1998).
[Crossref]

Qiu, L.

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, “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,” in Proc. Royal Soc. London A: Math. Phys. Eng. Sci., vol. 253 (The Royal Society, 1959), pp. 358–379.
[Crossref]

Rogach, A.

S. Götzinger, L. de S Menezes, O. Benson, D. Talapin, N. Gaponik, H. Weller, A. Rogach, and V. Sandoghdar, “Confocal microscopy and spectroscopy of nanocrystals on a high-q microsphere resonator,” J. Opt. B 6, 154 (2004).
[Crossref]

Roques, M.

S. Le Person, J. Puiggali, M. Baron, and M. Roques, “Near infrared drying of pharmaceutical thin films: experimental analysis of internal mass transport,” Chem. Eng. Process. Process. Intensif. 37, 257–263 (1998).
[Crossref]

Rust, M. J.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (storm),” Nat. Methods 3, 793 (2006).
[Crossref] [PubMed]

Sandoghdar, V.

S. Götzinger, L. de S Menezes, O. Benson, D. Talapin, N. Gaponik, H. Weller, A. Rogach, and V. Sandoghdar, “Confocal microscopy and spectroscopy of nanocrystals on a high-q microsphere resonator,” J. Opt. B 6, 154 (2004).
[Crossref]

K. Lindfors, T. Kalkbrenner, P. Stoller, and V. Sandoghdar, “Detection and spectroscopy of gold nanoparticles using supercontinuum white light confocal microscopy,” Phys. Rev. Lett. 93, 037401 (2004).
[Crossref] [PubMed]

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 (2008).
[Crossref]

Sheppard, C. J.

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 (2008).
[Crossref]

Sougrat, R.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref] [PubMed]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of mathematical functions: with formulas, graphs, and mathematical tables, vol. 55 (Courier Corporation).

Stoller, P.

K. Lindfors, T. Kalkbrenner, P. Stoller, and V. Sandoghdar, “Detection and spectroscopy of gold nanoparticles using supercontinuum white light confocal microscopy,” Phys. Rev. Lett. 93, 037401 (2004).
[Crossref] [PubMed]

Sui, G.

Sun, Y.

Talapin, D.

S. Götzinger, L. de S Menezes, O. Benson, D. Talapin, N. Gaponik, H. Weller, A. Rogach, and V. Sandoghdar, “Confocal microscopy and spectroscopy of nanocrystals on a high-q microsphere resonator,” J. Opt. B 6, 154 (2004).
[Crossref]

Tang, F.

Tang, W. T.

Török, P.

P. Török, P. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[Crossref]

Urbach, H.

K. Ushakova, Q. Van den Berg, S. Pereira, and H. Urbach, “Demonstration of spot size reduction by focussing amplitude modulated radially polarized light on a photoresist,” J. Opt. 17, 125615 (2015).
[Crossref]

H. Urbach and S. Pereira, “Field in focus with a maximum longitudinal electric component,” Phys. Rev. Lett. 100, 123904 (2008).
[Crossref] [PubMed]

Ushakova, K.

K. Ushakova, Q. Van den Berg, S. Pereira, and H. Urbach, “Demonstration of spot size reduction by focussing amplitude modulated radially polarized light on a photoresist,” J. Opt. 17, 125615 (2015).
[Crossref]

Van den Berg, Q.

K. Ushakova, Q. Van den Berg, S. Pereira, and H. Urbach, “Demonstration of spot size reduction by focussing amplitude modulated radially polarized light on a photoresist,” J. Opt. 17, 125615 (2015).
[Crossref]

Vieth, M.

A. Hoffman, M. Goetz, M. Vieth, P. Galle, M. Neurath, and R. Kiesslich, “Confocal laser endomicroscopy: technical status and current indications,” Endoscopy 38, 1275–1283 (2006).
[Crossref] [PubMed]

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 (2008).
[Crossref]

Wang, S.

Wang, Y.

Weller, H.

S. Götzinger, L. de S Menezes, O. Benson, D. Talapin, N. Gaponik, H. Weller, A. Rogach, and V. Sandoghdar, “Confocal microscopy and spectroscopy of nanocrystals on a high-q microsphere resonator,” J. Opt. B 6, 154 (2004).
[Crossref]

Weng, X.

Wichmann, J.

Wilson, T.

P. Török, P. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[Crossref]

T. Wilson, R. Juškaitis, and P. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes,” Opt. Commun. 141, 298 – 313 (1997).
[Crossref]

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. ii. structure of the image field in an aplanatic system,” in Proc. Royal Soc. London A: Math. Phys. Eng. Sci., vol. 253 (The Royal Society, 1959), pp. 358–379.
[Crossref]

E. Wolf, “Electromagnetic diffraction in optical systems. i. an integral representation of the image field,” in Proc. Royal Soc. London A: Math. Phys. Eng. Sci., vol. 253 (The Royal Society, 1959), pp. 349–357.
[Crossref]

Xie, X.

X. Xie, Y. Chen, K. Yang, and J. Zhou, “Harnessing the point-spread function for high-resolution far-field optical microscopy,” Phys. Rev. Lett. 113, 263901 (2014).
[Crossref]

L. Yang, X. Xie, S. Wang, and J. Zhou, “Minimized spot of annular radially polarized focusing beam,” Opt. Lett. 38, 1331–1333 (2013).
[Crossref] [PubMed]

Yang, K.

X. Xie, Y. Chen, K. Yang, and J. Zhou, “Harnessing the point-spread function for high-resolution far-field optical microscopy,” Phys. Rev. Lett. 113, 263901 (2014).
[Crossref]

Yang, L.

Yew, E. Y.

Youngworth, K. S.

Zhao, W.

Zhao, Y.

Zhou, J.

X. Xie, Y. Chen, K. Yang, and J. Zhou, “Harnessing the point-spread function for high-resolution far-field optical microscopy,” Phys. Rev. Lett. 113, 263901 (2014).
[Crossref]

L. Yang, X. Xie, S. Wang, and J. Zhou, “Minimized spot of annular radially polarized focusing beam,” Opt. Lett. 38, 1331–1333 (2013).
[Crossref] [PubMed]

Zhuang, S.

Zhuang, X.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (storm),” Nat. Methods 3, 793 (2006).
[Crossref] [PubMed]

Appl. Opt. (3)

Chem. Eng. Process. Process. Intensif. (1)

S. Le Person, J. Puiggali, M. Baron, and M. Roques, “Near infrared drying of pharmaceutical thin films: experimental analysis of internal mass transport,” Chem. Eng. Process. Process. Intensif. 37, 257–263 (1998).
[Crossref]

Clin. Exp. Ophthalmol. (1)

D. V. Patel and C. N. McGhee, “Contemporary in vivo confocal microscopy of the living human cornea using white light and laser scanning techniques: a major review,” Clin. Exp. Ophthalmol. 35, 71–88 (2007).
[Crossref] [PubMed]

Endoscopy (1)

A. Hoffman, M. Goetz, M. Vieth, P. Galle, M. Neurath, and R. Kiesslich, “Confocal laser endomicroscopy: technical status and current indications,” Endoscopy 38, 1275–1283 (2006).
[Crossref] [PubMed]

J. Mod. Opt. (1)

P. Török, P. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[Crossref]

J. Opt. (1)

K. Ushakova, Q. Van den Berg, S. Pereira, and H. Urbach, “Demonstration of spot size reduction by focussing amplitude modulated radially polarized light on a photoresist,” J. Opt. 17, 125615 (2015).
[Crossref]

J. Opt. B (1)

S. Götzinger, L. de S Menezes, O. Benson, D. Talapin, N. Gaponik, H. Weller, A. Rogach, and V. Sandoghdar, “Confocal microscopy and spectroscopy of nanocrystals on a high-q microsphere resonator,” J. Opt. B 6, 154 (2004).
[Crossref]

Nat. Methods (1)

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (storm),” Nat. Methods 3, 793 (2006).
[Crossref] [PubMed]

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 (2008).
[Crossref]

Opt. Commun. (2)

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

T. Wilson, R. Juškaitis, and P. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes,” Opt. Commun. 141, 298 – 313 (1997).
[Crossref]

Opt. Express (2)

Opt. Lett. (6)

Phys. Rev. Lett. (4)

K. Lindfors, T. Kalkbrenner, P. Stoller, and V. Sandoghdar, “Detection and spectroscopy of gold nanoparticles using supercontinuum white light confocal microscopy,” Phys. Rev. Lett. 93, 037401 (2004).
[Crossref] [PubMed]

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

X. Xie, Y. Chen, K. Yang, and J. Zhou, “Harnessing the point-spread function for high-resolution far-field optical microscopy,” Phys. Rev. Lett. 113, 263901 (2014).
[Crossref]

H. Urbach and S. Pereira, “Field in focus with a maximum longitudinal electric component,” Phys. Rev. Lett. 100, 123904 (2008).
[Crossref] [PubMed]

Science (1)

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref] [PubMed]

Trans. Opt. Inst (1)

V. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” Trans. Opt. Inst 1, 1–36 (1919).

Other (6)

E. Wolf, “Electromagnetic diffraction in optical systems. i. an integral representation of the image field,” in Proc. Royal Soc. London A: Math. Phys. Eng. Sci., vol. 253 (The Royal Society, 1959), pp. 349–357.
[Crossref]

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. ii. structure of the image field in an aplanatic system,” in Proc. Royal Soc. London A: Math. Phys. Eng. Sci., vol. 253 (The Royal Society, 1959), pp. 358–379.
[Crossref]

M. Abramowitz and I. A. Stegun, Handbook of mathematical functions: with formulas, graphs, and mathematical tables, vol. 55 (Courier Corporation).

J. Jackson, Classical Electrodynamics, 2nd ed.(John Wiley & Sons).

D. B. Murphy, Fundamentals of Light Microscopy and Electronic Imaging (John Wiley & Sons).

L. Novotny and B. Hecht, Principles of nano-optics (Cambridge University).

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Figures (6)

Fig. 1
Fig. 1 Schematic of the confocal microscope. A beam splitter divides the excitation path and detection path into two arms. A laser beam is focused onto the sample by a high NA objective lens L1. The light scattered by the sample is collected by the same objective lens and focused by a small NA lens onto a pinhole in front of a detector. a) The complete confocal microscopy system, b) Focusing and dipole excitation, c) Imaging and polarization conversion.
Fig. 2
Fig. 2 Profiles of the squared amplitude of the longitudinal and transverse components and total energy density | E | 2 in the focal plane in the case of a pupil field that is linearly polarized in the x–direction (a) and a pupil field that is radially polarized (b). The focusing lens has NA=0.9. The plots are normalized to the on-axis peak values.
Fig. 3
Fig. 3 Profiles of the squared amplitudes of the longitudinal components of the exciting spot in the focal plane in the case of radially polarized illumination with full aperture, annular aperture and optimized radially polarized pupil field. The plots are normalized to their on-axis maxima.
Fig. 4
Fig. 4 Profiles of the intensities at the detector plane with a polarization convertor and a small pinhole with radius r = 0.18µm which is 0.36λ in terms of wavelength. The system is composed of: a high NA1 = 0.9 focusing objective lens and a low NA2 = 0.3 lens for detection. For illumination, radially polarized light of wavelength λ = 500nm is used. The focal field can be seen in Eq. (17). The dipole is set rigidly along the z α–axis direction (only αzz is relevant). The plots are normalized to on-axis maxima.
Fig. 5
Fig. 5 Profiles of the detected intensities as function of scanning distance of two separated dipoles for different kinds of illumination and for several distances between the dipoles. The system is composed of two lenses with NA1 = 0.9, and NA2 = 0.3 and is illuminated by light with a wavelength of λ = 500nm. The illumination is either linearly, radially or optimized radially polarized. The two dipoles are set along the x axis for the linear polarization and along the z axis for the other radially polarized cases. A polarization convertor is added in the case of radially polarized excitation. Four different distances between the dipoles are chosen: d=0.8λ (red line), d=0.6λ (black line), d=0.4λ (yellow line), and d=0.36λ (blue line). The plots are normalized to the on-axis peak intensity.
Fig. 6
Fig. 6 Profiles of the visibility as a function of the d/λ for the four cases of pupil fields. The system is the same as shown in Fig. 5.

Equations (44)

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E e ( r ) = 1 4 π 2 k k 0 NA 1 A ( k ) exp ( i k r ) d 2 k , H e ( r ) = 1 4 π 2 1 ω μ 0 k k 0 NA 1 A × A ( k ) exp ( i k r ) d 2 k ,
k ^ = k x k x ^ + k y k y ^ + k z k z ^ p ^ ( k ) = k x k z x ^ + k y k z y ^ ( k x 2 + k y 2 ) z ^ k k x   2 + k y   2 , s ^ ( k ) = k y x ^ + k x y ^ k x   2 + k y   2 ,
A ( k ) = A p ( k ) p ^ ( k ) + A s ( k ) s ^ ( k ) ,
E e ( r ) = 1 4 π 2 k k 0 NA 1 ( A p p ^ + A s s ^ ) e i k r d 2 k , H e ( r ) = 1 4 π 2 1 ω μ 0 k k 0 NA 1 ( A s p ^ + A p s ^ ) e i k r d 2 k .
A p ( k ) = 2 π i f 1 k k z E ρ e , p ( ρ p , φ p ) , A s ( k ) = 2 π i f 1 k k z E φ e , p ( ρ p , φ p ) ,
ρ p = f 1 k k 0 n , ρ p cos φ p = f 1 k x k 0 n , ρ p sin φ p = f 1 k y k 0 n ,
cos φ p = k x / k , sin φ p = k y / k .
E ρ e , p ( ρ p , φ p ) = g ( ρ p ) cos φ p , E φ e , p ( ρ p , φ p ) = g ( ρ p ) sin φ p ,
A p ( k ) = 2 π i f 1 k x k k z k g ( k ) , A s ( k ) = 2 π i f 1 k y k k z k g ( k ) ,
0 2 π cos n ϕ e i x cos ( ϕ φ ) d ϕ = 2 π i n J n ( x ) cos n φ , 0 2 π sin n ϕ e i x cos ( ϕ φ ) d ϕ = 2 π i n J n ( x ) sin n φ ,
E e ( ρ , φ , z ) = i f 1 2 k 3 / 2 ( I 00 ( ρ , z ) I 02 ( ρ , z ) cos 2 φ I 02 ( ρ , z ) sin 2 φ 2 i I 01 ( ρ , z ) cos φ ) ,
I 00 ( ρ , z ) = 0 k 0 NA 1 g ( k ) ( k + k z ) J 0 ( k ρ ) k k z e i k z z d k ,
I 01 ( ρ , z ) = 0 k 0 NA 1 g ( k ) k J 1 ( k ρ ) k k z e i k z z d k ,
I 02 ( ρ , z ) = 0 k 0 NA 1 g ( k ) ( k k z ) J 2 ( k ρ ) k k z e i k z z d k .
E ρ e , p ( ρ p , φ p ) = g ( ρ p ) , E φ e , p ( ρ p , φ p ) = 0 ,
A p ( k ) = 2 π i f 1 k k z g ( k ) , A s ( k ) = 0 ,
E e ( ρ , φ , z ) = f 1 k 3 / 2 ( I 11 ( ρ , z ) cos φ I 11 ( ρ , z ) sin φ i I 10 ( ρ , z ) ) ,
I 10 ( ρ , z ) = 0 k 0 NA 1 g ( k ) k J 0 ( k ρ ) k k z e i k z z d k ,
I 11 ( ρ , z ) = 0 k 0 NA 1 g ( k ) k z J 1 ( k ρ ) k k z e i k z z d k .
g ( k ) = k 3 / 2 k 1 / 2 2 π i f 1 k z Λ ,
Λ = ( π P 0 ) 1 / 2 n 1 / 2 λ 0 ( ϵ 0 μ 0 ) 1 / 4 ( 2 3 1 ( NA 1 / n ) 2 + 1 3 ( 1 ( NA 1 / n ) 2 ) 3 ) 1 / 2 ,
I 10 ( ρ , z ) = k 1 / 2 2 π i f 1 Λ 0 k 0 NA 1 J 0 ( k ρ ) k 7 / 2 k z 3 / 2 e i k z z d k ,
I 11 ( ρ , z ) = k 1 / 2 2 π i f 1 Λ 0 k 0 NA 1 J 1 ( k ρ ) k 5 / 2 k z 3 / 2 e i k z z d k .
g ( ρ p ) = { 1 a Δ ρ p < ρ p a 0 otherwise ,
P d = α E e ( r d ) ,
α = ( α x x 0 0 0 α y y 0 0 0 α z z ) .
A d ( k ) = e i k z f 1 2 i ϵ 0 n 2 k z k × ( k × P d ) ,
A p d ( k ) = e i k z f 1 2 i ϵ 0 n 2 k z k 2 P d p ^ , A s d ( k ) = e i k z f 1 2 i ϵ 0 n 2 k z k 2 P d s ^ ,
E ρ d , p ( ρ p , φ p ) = k k z 2 π i f 1 A p d ( k ) , E φ d , p ( ρ p , φ p ) = k k z 2 π i f 1 A s d ( k ) .
A ˜ p d ( k ˜ ) = 2 π i f 2 k k ˜ z E ρ d , p ( ρ p , φ p ) = f 2 f 1 k z k ˜ z e i k z f 1 2 i ϵ 0 n 2 k z k 2 P d p ^ , A ˜ s d ( k ˜ ) = 2 π i f 2 k k ˜ z E φ d , p ( ρ p , φ p ) = f 2 f 1 k z k ˜ z e i k z f 1 2 i ϵ 0 n 2 k z k 2 P d s ^ ,
ρ p = f 2 k ˜ k , ρ p cos φ p = f 2 k ˜ x k , ρ p sin φ p = f 2 k ˜ y k ,
k = f 2 f 1 k ˜ , k z = k 2 ( f 2 / f 1 ) 2 k ˜ 2 .
E ( i ) ( r ) = 1 4 π 2 [ k ˜ k 0 NA 2 ( A ˜ p d ( k ˜ ) p ^ ( k ˜ ) + A ˜ s d ( k ˜ ) s ^ ( k ˜ ) ) e i k ˜ r d 2 k ˜ ] P d = k 2 f 2 4 π i ϵ 0 n 2 f 1 [ k ˜ k 0 NA 2 1 k z k ˜ z ( p ^ ( k ˜ ) p ^ ( f 2 f 1 k ˜ ) + s ^ ( k ˜ ) s ^ ( f 2 f 1 k ˜ ) ) e i k z f 1 e i k ˜ r d 2 k ˜ ] P d ,
p ^ ( k ˜ ) = 1 k 2 k ˜ ( k ˜ x k ˜ z k ˜ y k ˜ z k ˜ 2 ) , p ^ ( f 2 f 1 k ˜ ) = 1 k 2 k ˜ ( k ˜ x k z k ˜ y k z f 2 f 1 k ˜ 2 ) , s ^ ( k ˜ ) = s ^ ( f 2 f 1 k ˜ ) = 1 k 2 k ˜ ( k ˜ y k ˜ x 0 ) ,
E x ( i ) ( ρ , φ , z ) = f 2 2 ϵ 0 n 2 f 1 [ ( K 00 0 + K z z 0 ) p x d i p 2 i K z 1 cos φ p z d i p + ( K 00 2 K z z 2 ) ( cos 2 φ p x d i p + sin 2 φ p y d i p ) ] , E y ( i ) ( ρ , φ , z ) = f 2 2 ϵ 0 n 2 f 1 [ ( K 00 0 + K z z 0 ) p y d i p 2 i K z 1 sin φ p z d i p + ( K 00 2 K z z 2 ) ( sin 2 φ p x d i p cos 2 φ p y d i p ) ] , E z ( i ) ( ρ , φ , z ) = f 2 2 ϵ 0 n 2 f 1 [ 2 K 0 p z d i p 2 i K z z 1 ( cos φ p x d i p + sin φ p y d i p ) ] ,
K 00 n ( ρ , z ) = 0 k 0 NA 2 ( k ˜ z k ) 1 / 2 k ˜ k k 2 ( f 2 / f 1 ) 2 k ˜ 2 J n ( k ˜ ρ ) e i k ˜ z z e i z k 2 ( f 2 / f 1 ) 2 k ˜ 2 d k ˜ , K n ( ρ , z ) = 0 k 0 NA 2 ( k ˜ z k ) 1 / 2 k ˜ k k 2 ( f 2 / f 1 ) 2 k ˜ 2 q ( k ˜ ) J n ( k ˜ ρ ) e i k ˜ z z e i z k 2 ( f 2 / f 1 ) 2 k ˜ 2 d k ˜ , K z n ( ρ , z ) = 0 k 0 NA 2 ( k ˜ z k ) 1 / 2 k ˜ k k 2 ( f 2 / f 1 ) 2 k ˜ 2 q z ( k ˜ ) J n ( k ˜ ρ ) e i k ˜ z z e i z k 2 ( f 2 / f 1 ) 2 k ˜ 2 d k ˜ , K z z n ( ρ , z ) = 0 k 0 NA 2 ( k ˜ z k ) 1 / 2 k ˜ k k 2 ( f 2 / f 1 ) 2 k ˜ 2 q z z ( k ˜ ) J n ( k ˜ ρ ) e i k ˜ z z e i z k 2 ( f 2 / f 1 ) 2 k ˜ 2 d k ˜ ,
q ( k ˜ ) = k k ˜ k 2 , q z ( k ˜ ) = k k ˜ z k 2 , q z z ( k ˜ ) = k z k ˜ z k 2 .
E ( i ) , p ( ρ p , φ p ) = E ρ d , p ( ρ p , φ p ) x ^ ,
E ρ ( i ) , p ( ρ p , φ p ) = E ρ ( i ) , p ( ρ p , φ p ) cos ϕ , E φ ( i ) , p ( ρ p , φ p ) = E ρ d , p ( ρ p , φ p ) sin ϕ .
A ˜ p ( k ˜ ) = 2 π i f 2 k ˜ k ˜ z E ρ ( i ) , p ( ρ p , φ p ) , A ˜ s ( k ˜ ) = 2 π i f 2 k ˜ k ˜ z E φ ( i ) , p ( ρ p , φ p ) ,
E x ( i ) ( ρ , φ , z ) = f 2 2 ϵ 0 n 2 f 1 [ i 2 ( K z z 3 K z 3 ) ( cos 3 φ p x d i p + sin 3 φ p y d i p ) i 2 K z z 1 ( 3 cos φ p x d i p + sin φ p y d i p ) 1 2 K z 1 ( cos φ p x d i p + 3 sin φ p y d i p ) ( K z 2 K 2 ) cos 2 φ p z d i p + ( K z 0 + K 0 ) p z d i p ] , E y ( i ) ( ρ , φ , z ) = f 2 2 ϵ 0 n 2 f 1 [ i 2 ( K z z 3 K z 3 ) ( sin 3 φ p x d i p cos 3 φ p y d i p ) i 2 ( K z z 1 K z 1 ) ( sin φ p x d i p + cos φ p y d i p ) ( K z 2 K 2 ) sin 2 φ p z d i p ] , E z ( i ) ( ρ , φ , z ) = f 2 2 ϵ 0 n 2 f 1 [ K z 1 p x d i p + K z 2 ( cos 2 φ p x d i p + sin 2 φ p y d i p ) + 2 i K 1 cos φ p z d i p ] ,
K z n ( ρ , z ) = 0 k 0 NA 2 ( k ˜ z k ) 1 / 2 k ˜ k k 2 ( f 2 / f 1 ) 2 k ˜ 2 q z ( k ˜ ) J n ( k ˜ ρ ) e i k ˜ z z e i z k 2 ( f 2 / f 1 ) 2 k ˜ 2 d k ˜ , K z n ( ρ , z ) = 0 k 0 NA 2 ( k ˜ z k ) 1 / 2 k ˜ k k 2 ( f 2 / f 1 ) 2 k ˜ 2 q z ( k ˜ ) J n ( k ˜ ρ ) e i k ˜ z z e i z k 2 ( f 2 / f 1 ) 2 k ˜ 2 d k ˜ , K n ( ρ , z ) = 0 k 0 NA 2 ( k ˜ z k ) 1 / 2 k ˜ k k 2 ( f 2 / f 1 ) 2 k ˜ 2 q ( k ˜ ) J n ( k ˜ ρ ) e i k ˜ z z e i z k 2 ( f 2 / f 1 ) 2 k ˜ 2 d k ˜ ,
q z ( k ˜ ) = k z k ˜ k 2 , q ( k ˜ ) = k k , q z ( k ˜ ) = k z k .
visibility = I m a x I m i n I m a x + I m i n .

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