Abstract

A simple roadmap is established for the construction of the smallest three-dimensional (3D) isotropic focal spots. It is achieved in a 4Pi configuration by imposing a restriction/condition of equal transverse and longitudinal spot sizes to determine the position of an annular aperture and then optimize its size. The calculations were performed for cylindrically symmetric radial, azimuthal, and circular polarizations for the cases of in-phase and out-of-phase counter-propagating beams as well as when a vortex was added to the beams. A diffraction-limited bright 3D isotropic spot containing solely longitudinal or transverse electric field components is obtained, while the 3D dark spot can be formed from one of two complementary combinations, each containing both transverse and longitudinal field components.

© 2013 Optical Society of America

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References

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  1. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009).
    [CrossRef]
  2. N. Bokor, N. Davidson, “4π focusing with single parabolic mirror,” Opt. Commun. 281, 5499–5503 (2008).
    [CrossRef]
  3. W. Chen, Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265, 411–417 (2006).
    [CrossRef]
  4. N. Bokor, N. Davidson, “A three dimensional dark focal spot uniformly surrounded by light,” Opt. Commun. 279, 229–234 (2007).
    [CrossRef]
  5. Y. Kozawa, S. Sato, “Dark spot formation by vector beams,” Opt. Lett. 33, 2326–2328 (2008).
    [CrossRef]
  6. X. L. Wang, J. Ding, J. Q. Qin, J. Chen, Y. X. Fan, H. T. Wang, “Configurable three-dimensional optical cage generated from cylindrical beams,” Opt. Commun. 282, 3421–3425 (2009).
    [CrossRef]
  7. N. Bokor, N. Davidson, “Toward a spherical spot distribution with 4π focusing of radially polarized light,” Opt. Lett. 29, 1968–1970 (2004).
    [CrossRef]
  8. N. Bokor, N. Davidson, “Generation of a hollow dark spherical spot by 4π focusing of a radially polarized Laguerre–Gaussian beam,” Opt. Lett. 31, 149–151 (2006).
    [CrossRef]
  9. Z. Chen, D. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. 37, 1286–1288 (2012).
    [CrossRef]
  10. S. Quabis, R. Dorn, M. Eberler, O. Glockl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
    [CrossRef]
  11. W. Chen, Q. Zhan, “Creating a spherical focal spot with spatially modulated radial polarization in 4Pi microscopy,” Opt. Lett. 34, 2444–2446 (2009).
    [CrossRef]
  12. W. Chen, Q. Zhan, “Three dimensional polarization control in 4Pi microscopy,” Opt. Commun. 284, 52–56 (2011).
    [CrossRef]
  13. J. Wang, W. Chen, Q. Zhan, “Creation of uniform three-dimensional optical chain through tight focusing of space-variant polarized beams,” J. Opt. 14, 055004 (2012).
    [CrossRef]
  14. E. Mudry, E. Le Moal, P. Ferrand, P. C. Chaumet, A. Sentenac, “Isotropic diffraction-limited focusing using a single objective lens,” Phys. Rev. Lett. 105, 203903 (2010).
    [CrossRef]
  15. M. Jang, A. Sentenac, C. Yang, “Optical phase conjugation (OPC)-assisted isotropic focusing,” Opt. Express 21, 8781–8792 (2013).
    [CrossRef]
  16. M. C. Lang, T. Staudt, J. Engelhardt, S. W. Hell, “4Pi microscopy with negligible sidelobes,” New J. Phys. 10, 043041 (2008).
    [CrossRef]
  17. V. Kalosha, I. Golub, “Toward the subdiffraction focusing limit of optical superresolution,” Opt. Lett. 32, 3540–3542 (2007).
    [CrossRef]
  18. S. N. Khonina, I. Golub, “Optimization of focusing of linearly polarized light,” Opt. Lett. 36, 352–354 (2011).
    [CrossRef]
  19. S. N. Khonina, I. Golub, “Enlightening darkness to diffraction limit and beyond: comparison and optimization of different polarizations for dark spot generation,” J. Opt. Soc. Am. A 29, 1470–1474 (2012).
    [CrossRef]
  20. T. Grosjean, D. Courjon, “Photopolymers as vectorial sensors of the electric field,” Opt. Express 14, 2203–2210 (2006).
    [CrossRef]
  21. T. Grosjean, D. Courjon, C. Bainier, “Smallest lithographic marks generated by optical focusing systems,” Opt. Lett. 32, 976–978 (2007).
    [CrossRef]
  22. S. N. Khonina, I. Golub, “How low can STED go? Comparison of different write-erase beam combinations for stimulated emission depletion microscopy,” J. Opt. Soc. Am. A 29, 2242–2246 (2012).
    [CrossRef]
  23. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the aplanatic system,” Proc. R. Soc. Lond. Ser. A 253, 358–379 (1959).
    [CrossRef]
  24. M. Dyba, S. W. Hell, “Focal spots of size λ/23 open up far-field fluorescence microscopy at 33 nm axial resolution,” Phys. Rev. Lett. 88, 163901 (2002).
    [CrossRef]
  25. J. Bewersdorf, A. Egner, S. W. Hell, Handbook of Biological Confocal Microscopy (Springer, 2006), p. 561.
  26. T. G. Kolda, M. R. Lewis, V. Torczon, “Optimization by direct search: new perspectives on some classical and modern methods,” SIAM Review 45, 385–482 (2003).
    [CrossRef]

2013 (1)

2012 (4)

S. N. Khonina, I. Golub, “How low can STED go? Comparison of different write-erase beam combinations for stimulated emission depletion microscopy,” J. Opt. Soc. Am. A 29, 2242–2246 (2012).
[CrossRef]

J. Wang, W. Chen, Q. Zhan, “Creation of uniform three-dimensional optical chain through tight focusing of space-variant polarized beams,” J. Opt. 14, 055004 (2012).
[CrossRef]

Z. Chen, D. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. 37, 1286–1288 (2012).
[CrossRef]

S. N. Khonina, I. Golub, “Enlightening darkness to diffraction limit and beyond: comparison and optimization of different polarizations for dark spot generation,” J. Opt. Soc. Am. A 29, 1470–1474 (2012).
[CrossRef]

2011 (2)

W. Chen, Q. Zhan, “Three dimensional polarization control in 4Pi microscopy,” Opt. Commun. 284, 52–56 (2011).
[CrossRef]

S. N. Khonina, I. Golub, “Optimization of focusing of linearly polarized light,” Opt. Lett. 36, 352–354 (2011).
[CrossRef]

2010 (1)

E. Mudry, E. Le Moal, P. Ferrand, P. C. Chaumet, A. Sentenac, “Isotropic diffraction-limited focusing using a single objective lens,” Phys. Rev. Lett. 105, 203903 (2010).
[CrossRef]

2009 (3)

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

X. L. Wang, J. Ding, J. Q. Qin, J. Chen, Y. X. Fan, H. T. Wang, “Configurable three-dimensional optical cage generated from cylindrical beams,” Opt. Commun. 282, 3421–3425 (2009).
[CrossRef]

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

2008 (3)

Y. Kozawa, S. Sato, “Dark spot formation by vector beams,” Opt. Lett. 33, 2326–2328 (2008).
[CrossRef]

N. Bokor, N. Davidson, “4π focusing with single parabolic mirror,” Opt. Commun. 281, 5499–5503 (2008).
[CrossRef]

M. C. Lang, T. Staudt, J. Engelhardt, S. W. Hell, “4Pi microscopy with negligible sidelobes,” New J. Phys. 10, 043041 (2008).
[CrossRef]

2007 (3)

V. Kalosha, I. Golub, “Toward the subdiffraction focusing limit of optical superresolution,” Opt. Lett. 32, 3540–3542 (2007).
[CrossRef]

N. Bokor, N. Davidson, “A three dimensional dark focal spot uniformly surrounded by light,” Opt. Commun. 279, 229–234 (2007).
[CrossRef]

T. Grosjean, D. Courjon, C. Bainier, “Smallest lithographic marks generated by optical focusing systems,” Opt. Lett. 32, 976–978 (2007).
[CrossRef]

2006 (3)

2004 (1)

2003 (1)

T. G. Kolda, M. R. Lewis, V. Torczon, “Optimization by direct search: new perspectives on some classical and modern methods,” SIAM Review 45, 385–482 (2003).
[CrossRef]

2002 (1)

M. Dyba, S. W. Hell, “Focal spots of size λ/23 open up far-field fluorescence microscopy at 33 nm axial resolution,” Phys. Rev. Lett. 88, 163901 (2002).
[CrossRef]

2000 (1)

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

1959 (1)

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the aplanatic system,” Proc. R. Soc. Lond. Ser. A 253, 358–379 (1959).
[CrossRef]

Bainier, C.

T. Grosjean, D. Courjon, C. Bainier, “Smallest lithographic marks generated by optical focusing systems,” Opt. Lett. 32, 976–978 (2007).
[CrossRef]

Bewersdorf, J.

J. Bewersdorf, A. Egner, S. W. Hell, Handbook of Biological Confocal Microscopy (Springer, 2006), p. 561.

Bokor, N.

N. Bokor, N. Davidson, “4π focusing with single parabolic mirror,” Opt. Commun. 281, 5499–5503 (2008).
[CrossRef]

N. Bokor, N. Davidson, “A three dimensional dark focal spot uniformly surrounded by light,” Opt. Commun. 279, 229–234 (2007).
[CrossRef]

N. Bokor, N. Davidson, “Generation of a hollow dark spherical spot by 4π focusing of a radially polarized Laguerre–Gaussian beam,” Opt. Lett. 31, 149–151 (2006).
[CrossRef]

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

Chaumet, P. C.

E. Mudry, E. Le Moal, P. Ferrand, P. C. Chaumet, A. Sentenac, “Isotropic diffraction-limited focusing using a single objective lens,” Phys. Rev. Lett. 105, 203903 (2010).
[CrossRef]

Chen, J.

X. L. Wang, J. Ding, J. Q. Qin, J. Chen, Y. X. Fan, H. T. Wang, “Configurable three-dimensional optical cage generated from cylindrical beams,” Opt. Commun. 282, 3421–3425 (2009).
[CrossRef]

Chen, W.

J. Wang, W. Chen, Q. Zhan, “Creation of uniform three-dimensional optical chain through tight focusing of space-variant polarized beams,” J. Opt. 14, 055004 (2012).
[CrossRef]

W. Chen, Q. Zhan, “Three dimensional polarization control in 4Pi microscopy,” Opt. Commun. 284, 52–56 (2011).
[CrossRef]

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

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

Chen, Z.

Z. Chen, D. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. 37, 1286–1288 (2012).
[CrossRef]

Courjon, D.

T. Grosjean, D. Courjon, C. Bainier, “Smallest lithographic marks generated by optical focusing systems,” Opt. Lett. 32, 976–978 (2007).
[CrossRef]

T. Grosjean, D. Courjon, “Photopolymers as vectorial sensors of the electric field,” Opt. Express 14, 2203–2210 (2006).
[CrossRef]

Davidson, N.

N. Bokor, N. Davidson, “4π focusing with single parabolic mirror,” Opt. Commun. 281, 5499–5503 (2008).
[CrossRef]

N. Bokor, N. Davidson, “A three dimensional dark focal spot uniformly surrounded by light,” Opt. Commun. 279, 229–234 (2007).
[CrossRef]

N. Bokor, N. Davidson, “Generation of a hollow dark spherical spot by 4π focusing of a radially polarized Laguerre–Gaussian beam,” Opt. Lett. 31, 149–151 (2006).
[CrossRef]

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

Ding, J.

X. L. Wang, J. Ding, J. Q. Qin, J. Chen, Y. X. Fan, H. T. Wang, “Configurable three-dimensional optical cage generated from cylindrical beams,” Opt. Commun. 282, 3421–3425 (2009).
[CrossRef]

Dorn, R.

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

Dyba, M.

M. Dyba, S. W. Hell, “Focal spots of size λ/23 open up far-field fluorescence microscopy at 33 nm axial resolution,” Phys. Rev. Lett. 88, 163901 (2002).
[CrossRef]

Eberler, M.

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

Egner, A.

J. Bewersdorf, A. Egner, S. W. Hell, Handbook of Biological Confocal Microscopy (Springer, 2006), p. 561.

Engelhardt, J.

M. C. Lang, T. Staudt, J. Engelhardt, S. W. Hell, “4Pi microscopy with negligible sidelobes,” New J. Phys. 10, 043041 (2008).
[CrossRef]

Fan, Y. X.

X. L. Wang, J. Ding, J. Q. Qin, J. Chen, Y. X. Fan, H. T. Wang, “Configurable three-dimensional optical cage generated from cylindrical beams,” Opt. Commun. 282, 3421–3425 (2009).
[CrossRef]

Ferrand, P.

E. Mudry, E. Le Moal, P. Ferrand, P. C. Chaumet, A. Sentenac, “Isotropic diffraction-limited focusing using a single objective lens,” Phys. Rev. Lett. 105, 203903 (2010).
[CrossRef]

Glockl, O.

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

Golub, I.

S. N. Khonina, I. Golub, “How low can STED go? Comparison of different write-erase beam combinations for stimulated emission depletion microscopy,” J. Opt. Soc. Am. A 29, 2242–2246 (2012).
[CrossRef]

S. N. Khonina, I. Golub, “Enlightening darkness to diffraction limit and beyond: comparison and optimization of different polarizations for dark spot generation,” J. Opt. Soc. Am. A 29, 1470–1474 (2012).
[CrossRef]

S. N. Khonina, I. Golub, “Optimization of focusing of linearly polarized light,” Opt. Lett. 36, 352–354 (2011).
[CrossRef]

V. Kalosha, I. Golub, “Toward the subdiffraction focusing limit of optical superresolution,” Opt. Lett. 32, 3540–3542 (2007).
[CrossRef]

Grosjean, T.

T. Grosjean, D. Courjon, C. Bainier, “Smallest lithographic marks generated by optical focusing systems,” Opt. Lett. 32, 976–978 (2007).
[CrossRef]

T. Grosjean, D. Courjon, “Photopolymers as vectorial sensors of the electric field,” Opt. Express 14, 2203–2210 (2006).
[CrossRef]

Hell, S. W.

M. C. Lang, T. Staudt, J. Engelhardt, S. W. Hell, “4Pi microscopy with negligible sidelobes,” New J. Phys. 10, 043041 (2008).
[CrossRef]

M. Dyba, S. W. Hell, “Focal spots of size λ/23 open up far-field fluorescence microscopy at 33 nm axial resolution,” Phys. Rev. Lett. 88, 163901 (2002).
[CrossRef]

J. Bewersdorf, A. Egner, S. W. Hell, Handbook of Biological Confocal Microscopy (Springer, 2006), p. 561.

Jang, M.

Kalosha, V.

V. Kalosha, I. Golub, “Toward the subdiffraction focusing limit of optical superresolution,” Opt. Lett. 32, 3540–3542 (2007).
[CrossRef]

Khonina, S. N.

Kolda, T. G.

T. G. Kolda, M. R. Lewis, V. Torczon, “Optimization by direct search: new perspectives on some classical and modern methods,” SIAM Review 45, 385–482 (2003).
[CrossRef]

Kozawa, Y.

Lang, M. C.

M. C. Lang, T. Staudt, J. Engelhardt, S. W. Hell, “4Pi microscopy with negligible sidelobes,” New J. Phys. 10, 043041 (2008).
[CrossRef]

Le Moal, E.

E. Mudry, E. Le Moal, P. Ferrand, P. C. Chaumet, A. Sentenac, “Isotropic diffraction-limited focusing using a single objective lens,” Phys. Rev. Lett. 105, 203903 (2010).
[CrossRef]

Leuchs, G.

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

Lewis, M. R.

T. G. Kolda, M. R. Lewis, V. Torczon, “Optimization by direct search: new perspectives on some classical and modern methods,” SIAM Review 45, 385–482 (2003).
[CrossRef]

Mudry, E.

E. Mudry, E. Le Moal, P. Ferrand, P. C. Chaumet, A. Sentenac, “Isotropic diffraction-limited focusing using a single objective lens,” Phys. Rev. Lett. 105, 203903 (2010).
[CrossRef]

Qin, J. Q.

X. L. Wang, J. Ding, J. Q. Qin, J. Chen, Y. X. Fan, H. T. Wang, “Configurable three-dimensional optical cage generated from cylindrical beams,” Opt. Commun. 282, 3421–3425 (2009).
[CrossRef]

Quabis, S.

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

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the aplanatic system,” Proc. R. Soc. Lond. Ser. A 253, 358–379 (1959).
[CrossRef]

Sato, S.

Sentenac, A.

M. Jang, A. Sentenac, C. Yang, “Optical phase conjugation (OPC)-assisted isotropic focusing,” Opt. Express 21, 8781–8792 (2013).
[CrossRef]

E. Mudry, E. Le Moal, P. Ferrand, P. C. Chaumet, A. Sentenac, “Isotropic diffraction-limited focusing using a single objective lens,” Phys. Rev. Lett. 105, 203903 (2010).
[CrossRef]

Staudt, T.

M. C. Lang, T. Staudt, J. Engelhardt, S. W. Hell, “4Pi microscopy with negligible sidelobes,” New J. Phys. 10, 043041 (2008).
[CrossRef]

Torczon, V.

T. G. Kolda, M. R. Lewis, V. Torczon, “Optimization by direct search: new perspectives on some classical and modern methods,” SIAM Review 45, 385–482 (2003).
[CrossRef]

Wang, H. T.

X. L. Wang, J. Ding, J. Q. Qin, J. Chen, Y. X. Fan, H. T. Wang, “Configurable three-dimensional optical cage generated from cylindrical beams,” Opt. Commun. 282, 3421–3425 (2009).
[CrossRef]

Wang, J.

J. Wang, W. Chen, Q. Zhan, “Creation of uniform three-dimensional optical chain through tight focusing of space-variant polarized beams,” J. Opt. 14, 055004 (2012).
[CrossRef]

Wang, X. L.

X. L. Wang, J. Ding, J. Q. Qin, J. Chen, Y. X. Fan, H. T. Wang, “Configurable three-dimensional optical cage generated from cylindrical beams,” Opt. Commun. 282, 3421–3425 (2009).
[CrossRef]

Wolf, E.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the aplanatic system,” Proc. R. Soc. Lond. Ser. A 253, 358–379 (1959).
[CrossRef]

Yang, C.

Zhan, Q.

J. Wang, W. Chen, Q. Zhan, “Creation of uniform three-dimensional optical chain through tight focusing of space-variant polarized beams,” J. Opt. 14, 055004 (2012).
[CrossRef]

W. Chen, Q. Zhan, “Three dimensional polarization control in 4Pi microscopy,” Opt. Commun. 284, 52–56 (2011).
[CrossRef]

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

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

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

Zhao, D.

Z. Chen, D. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. 37, 1286–1288 (2012).
[CrossRef]

Adv. Opt. Photon. (1)

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

J. Opt. (1)

J. Wang, W. Chen, Q. Zhan, “Creation of uniform three-dimensional optical chain through tight focusing of space-variant polarized beams,” J. Opt. 14, 055004 (2012).
[CrossRef]

J. Opt. Soc. Am. A (1)

S. N. Khonina, I. Golub, “Enlightening darkness to diffraction limit and beyond: comparison and optimization of different polarizations for dark spot generation,” J. Opt. Soc. Am. A 29, 1470–1474 (2012).
[CrossRef]

J. Opt. Soc. Am. A (1)

New J. Phys. (1)

M. C. Lang, T. Staudt, J. Engelhardt, S. W. Hell, “4Pi microscopy with negligible sidelobes,” New J. Phys. 10, 043041 (2008).
[CrossRef]

Opt. Commun. (3)

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

N. Bokor, N. Davidson, “A three dimensional dark focal spot uniformly surrounded by light,” Opt. Commun. 279, 229–234 (2007).
[CrossRef]

X. L. Wang, J. Ding, J. Q. Qin, J. Chen, Y. X. Fan, H. T. Wang, “Configurable three-dimensional optical cage generated from cylindrical beams,” Opt. Commun. 282, 3421–3425 (2009).
[CrossRef]

Opt. Lett. (3)

Z. Chen, D. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. 37, 1286–1288 (2012).
[CrossRef]

V. Kalosha, I. Golub, “Toward the subdiffraction focusing limit of optical superresolution,” Opt. Lett. 32, 3540–3542 (2007).
[CrossRef]

T. Grosjean, D. Courjon, C. Bainier, “Smallest lithographic marks generated by optical focusing systems,” Opt. Lett. 32, 976–978 (2007).
[CrossRef]

Opt. Commun. (3)

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

W. Chen, Q. Zhan, “Three dimensional polarization control in 4Pi microscopy,” Opt. Commun. 284, 52–56 (2011).
[CrossRef]

N. Bokor, N. Davidson, “4π focusing with single parabolic mirror,” Opt. Commun. 281, 5499–5503 (2008).
[CrossRef]

Opt. Express (2)

Opt. Lett. (5)

Phys. Rev. Lett. (1)

E. Mudry, E. Le Moal, P. Ferrand, P. C. Chaumet, A. Sentenac, “Isotropic diffraction-limited focusing using a single objective lens,” Phys. Rev. Lett. 105, 203903 (2010).
[CrossRef]

Phys. Rev. Lett. (1)

M. Dyba, S. W. Hell, “Focal spots of size λ/23 open up far-field fluorescence microscopy at 33 nm axial resolution,” Phys. Rev. Lett. 88, 163901 (2002).
[CrossRef]

Proc. R. Soc. Lond. Ser. A (1)

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the aplanatic system,” Proc. R. Soc. Lond. Ser. A 253, 358–379 (1959).
[CrossRef]

SIAM Review (1)

T. G. Kolda, M. R. Lewis, V. Torczon, “Optimization by direct search: new perspectives on some classical and modern methods,” SIAM Review 45, 385–482 (2003).
[CrossRef]

Other (1)

J. Bewersdorf, A. Egner, S. W. Hell, Handbook of Biological Confocal Microscopy (Springer, 2006), p. 561.

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Figures (1)

Fig. 1.
Fig. 1.

Intensity distribution for radial incident polarization in the focal plane (a) and (c) and along the optical axis (b) and (d) for an annular aperture placed at r c = 83 λ and having a width Δ = λ (a) and (b) and Δ = 10 λ (c) and (d).

Tables (1)

Tables Icon

Table 1. Transverse and On-Optical Axis Intensity Distribution ( 4 λ × 4 λ ) in the Focus for Different Cases

Equations (9)

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

E z ( ρ , φ , z ) = i k f α 1 α 2 cos θ exp ( i k z cos θ ) J 0 ( k ρ sin θ ) sin 2 θ d θ ,
E z ( ρ ) i k f ( α 2 α 1 ) cos α c sin 2 ( α c ) J 0 ( k ρ sin α c )
E z ( z ) f z cos α c sin ( α c ) [ exp ( i k z cos α 1 ) exp ( i k z cos α 2 ) ] .
ρ 0 = 2.4 k sin α c ,
z 0 = π k sin α c sin ( Δ / 2 ) ,
E 4 p i ( r , φ , z ) = E 1 ( r , φ , z ) + exp ( i δ ) E 2 ( r , φ , z ) .
E z ( z ) = i k f α 1 α 2 cos ( k z cos θ ) cos θ sin 2 θ d θ i k f ( α 2 α 1 ) cos α c sin 2 ( α c ) cos ( k z cos α c ) .
z 0 = π 2 k cos α c .
tan α c = 4.8 π α c = 56.8 ° .

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