Abstract

Effects of primary aberrations including spherical aberration, coma and astigmatism on the three fluorescence depletion patterns mainly used in stimulated emission of depletion (STED) microscopy are investigated by using vectorial integral. The three depletion patterns are created by inserting a vortex phase plate, a central half-wavelength plate or a semi-circular half-wavelength mask within Gaussian beam respectively. Attention is given to the modification of the shape, peak intensity, the central intensity of the dark hole and the hole size of these depletion patterns in the presence of primary aberrations.

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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(11), 780–782 (1994).
    [CrossRef] [PubMed]
  2. S. W. Hell, “Far-Field Optical Nanoscopy,” Science 316(5828), 1153–1158 (2007).
    [CrossRef] [PubMed]
  3. S. W. Hell, “Microscopy and its focal switch,” Nat. Methods 6(1), 24–32 (2009).
    [CrossRef] [PubMed]
  4. T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(6), 066613 (2001).
    [CrossRef] [PubMed]
  5. T. Watanabe, M. Fujii, Y. Watanabe, N. Toyama, and Y. Iketaki, “Generation of a doughnut-shaped beam using a spiral phase plate,” Rev. Sci. Instrum. 75(12), 5131–5135 (2004).
    [CrossRef]
  6. J. Keller, A. Schönle, and S. W. Hell, “Efficient fluorescence inhibition patterns for RESOLFT microscopy,” Opt. Express 15(6), 3361–3371 (2007).
    [CrossRef] [PubMed]
  7. K. I. Willig, J. Keller, M. Bossi, and S. W. Hell, “STED microscopy resolves nanoparticle assemblies,” N. J. Phys. 8(6), 106 (2006).
    [CrossRef]
  8. P. Török and P. R. T. Munro, “The use of Gauss-Laguerre vector beams in STED microscopy,” Opt. Express 12(15), 3605–3617 (2004).
    [CrossRef] [PubMed]
  9. D. P. Biss and T. G. Brown, “Primary aberrations in focused radially polarized vortex beams,” Opt. Express 12(3), 384–393 (2004).
    [CrossRef] [PubMed]
  10. B. R. Boruah and M. A. A. Neil, “Susceptibility to and correction of azimuthal aberrations in singular light beams,” Opt. Express 14(22), 10377–10385 (2006).
    [CrossRef] [PubMed]
  11. B. R. Boruah and M. A. A. Neil, “Far field computation of an arbitrarily polarized beam using fast Fourier thrnsforms,” Opt. Commun. 282(24), 4660–4667 (2009).
    [CrossRef]
  12. M. Born, and E. Wolf, Principles of Optics, (Cambridge University Press, Cambridge, UK, 1999).
  13. R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations I. Spherical aberration, curvature of field, and distortion,” J. Mod. Opt. 40(11), 2293–2310 (1993).
    [CrossRef]
  14. R. Kant, “An analytical method of vector diffraction for focusing optical systems with Seidel aberrations II: Astigmatism and coma,” J. Mod. Opt. 42(2), 299–320 (1995).
    [CrossRef]
  15. R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary spherical aberration on high-numericalaperture focusing of a Laguerre-Gaussian beam,” J. Opt. Soc. Am. A 25(6), 1307–1318 (2008).
    [CrossRef]
  16. R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of a vortex carrying beam with Gaussian background by an apertured system in presence of coma,” Opt. Commun. 281(5), 923–934 (2008).
    [CrossRef]
  17. R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of linearly-, and circularly polarized Gaussian background vortex beams by a high numerical aperture system afflicted with third-order astigmatism,” Opt. Commun. 281(24), 5939–5948 (2008).
    [CrossRef]
  18. S. H. Deng, L. Liu, Y. Cheng, R. X. Li, and Z. Z. Xu, “Investigation of the influence of the aberration induced by a plane interface on STED microscopy,” Opt. Express 17(3), 1714–1725 (2009).
    [CrossRef] [PubMed]
  19. Y. Roichman, A. Waldron, E. Gardel, and D. G. Grier, “Optical traps with geometric aberrations,” Appl. Opt. 45(15), 3425–3429 (2006).
    [CrossRef] [PubMed]
  20. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II: Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
    [CrossRef]
  21. V. Westphal and S. W. Hell, “Nanoscale resolution in the focal plane of an optical microscope,” Phys. Rev. Lett. 94(14), 143903 (2005).
    [CrossRef] [PubMed]
  22. P. Dedecker, B. Muls, J. Hofkens, J. Enderlein, and J. Hotta, “Orientational effects in the excitation and de-excitation of single molecules interacting with donut-mode laser beams,” Opt. Express 15(6), 3372–3383 (2007).
    [CrossRef] [PubMed]
  23. N. Bokor, Y. Iketaki, T. Watanabe, K. Daigoku, N. Davidson, and M. Fujii, “On polarization effects in fluorescence depletion microscopy,” Opt. Commun. 272(1), 263–268 (2007).
    [CrossRef]

2009

S. W. Hell, “Microscopy and its focal switch,” Nat. Methods 6(1), 24–32 (2009).
[CrossRef] [PubMed]

B. R. Boruah and M. A. A. Neil, “Far field computation of an arbitrarily polarized beam using fast Fourier thrnsforms,” Opt. Commun. 282(24), 4660–4667 (2009).
[CrossRef]

S. H. Deng, L. Liu, Y. Cheng, R. X. Li, and Z. Z. Xu, “Investigation of the influence of the aberration induced by a plane interface on STED microscopy,” Opt. Express 17(3), 1714–1725 (2009).
[CrossRef] [PubMed]

2008

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary spherical aberration on high-numericalaperture focusing of a Laguerre-Gaussian beam,” J. Opt. Soc. Am. A 25(6), 1307–1318 (2008).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of a vortex carrying beam with Gaussian background by an apertured system in presence of coma,” Opt. Commun. 281(5), 923–934 (2008).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of linearly-, and circularly polarized Gaussian background vortex beams by a high numerical aperture system afflicted with third-order astigmatism,” Opt. Commun. 281(24), 5939–5948 (2008).
[CrossRef]

2007

2006

2005

V. Westphal and S. W. Hell, “Nanoscale resolution in the focal plane of an optical microscope,” Phys. Rev. Lett. 94(14), 143903 (2005).
[CrossRef] [PubMed]

2004

2001

T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(6), 066613 (2001).
[CrossRef] [PubMed]

1995

R. Kant, “An analytical method of vector diffraction for focusing optical systems with Seidel aberrations II: Astigmatism and coma,” J. Mod. Opt. 42(2), 299–320 (1995).
[CrossRef]

1994

1993

R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations I. Spherical aberration, curvature of field, and distortion,” J. Mod. Opt. 40(11), 2293–2310 (1993).
[CrossRef]

1959

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

Biss, D. P.

Bokor, N.

N. Bokor, Y. Iketaki, T. Watanabe, K. Daigoku, N. Davidson, and M. Fujii, “On polarization effects in fluorescence depletion microscopy,” Opt. Commun. 272(1), 263–268 (2007).
[CrossRef]

Boruah, B. R.

B. R. Boruah and M. A. A. Neil, “Far field computation of an arbitrarily polarized beam using fast Fourier thrnsforms,” Opt. Commun. 282(24), 4660–4667 (2009).
[CrossRef]

B. R. Boruah and M. A. A. Neil, “Susceptibility to and correction of azimuthal aberrations in singular light beams,” Opt. Express 14(22), 10377–10385 (2006).
[CrossRef] [PubMed]

Bossi, M.

K. I. Willig, J. Keller, M. Bossi, and S. W. Hell, “STED microscopy resolves nanoparticle assemblies,” N. J. Phys. 8(6), 106 (2006).
[CrossRef]

Brown, T. G.

Cheng, Y.

Daigoku, K.

N. Bokor, Y. Iketaki, T. Watanabe, K. Daigoku, N. Davidson, and M. Fujii, “On polarization effects in fluorescence depletion microscopy,” Opt. Commun. 272(1), 263–268 (2007).
[CrossRef]

Davidson, N.

N. Bokor, Y. Iketaki, T. Watanabe, K. Daigoku, N. Davidson, and M. Fujii, “On polarization effects in fluorescence depletion microscopy,” Opt. Commun. 272(1), 263–268 (2007).
[CrossRef]

Dedecker, P.

Deng, S. H.

Enderlein, J.

Engel, E.

T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(6), 066613 (2001).
[CrossRef] [PubMed]

Fujii, M.

N. Bokor, Y. Iketaki, T. Watanabe, K. Daigoku, N. Davidson, and M. Fujii, “On polarization effects in fluorescence depletion microscopy,” Opt. Commun. 272(1), 263–268 (2007).
[CrossRef]

T. Watanabe, M. Fujii, Y. Watanabe, N. Toyama, and Y. Iketaki, “Generation of a doughnut-shaped beam using a spiral phase plate,” Rev. Sci. Instrum. 75(12), 5131–5135 (2004).
[CrossRef]

Gardel, E.

Grier, D. G.

Hell, S. W.

S. W. Hell, “Microscopy and its focal switch,” Nat. Methods 6(1), 24–32 (2009).
[CrossRef] [PubMed]

J. Keller, A. Schönle, and S. W. Hell, “Efficient fluorescence inhibition patterns for RESOLFT microscopy,” Opt. Express 15(6), 3361–3371 (2007).
[CrossRef] [PubMed]

S. W. Hell, “Far-Field Optical Nanoscopy,” Science 316(5828), 1153–1158 (2007).
[CrossRef] [PubMed]

K. I. Willig, J. Keller, M. Bossi, and S. W. Hell, “STED microscopy resolves nanoparticle assemblies,” N. J. Phys. 8(6), 106 (2006).
[CrossRef]

V. Westphal and S. W. Hell, “Nanoscale resolution in the focal plane of an optical microscope,” Phys. Rev. Lett. 94(14), 143903 (2005).
[CrossRef] [PubMed]

T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(6), 066613 (2001).
[CrossRef] [PubMed]

S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. 19(11), 780–782 (1994).
[CrossRef] [PubMed]

Hofkens, J.

Hotta, J.

Iketaki, Y.

N. Bokor, Y. Iketaki, T. Watanabe, K. Daigoku, N. Davidson, and M. Fujii, “On polarization effects in fluorescence depletion microscopy,” Opt. Commun. 272(1), 263–268 (2007).
[CrossRef]

T. Watanabe, M. Fujii, Y. Watanabe, N. Toyama, and Y. Iketaki, “Generation of a doughnut-shaped beam using a spiral phase plate,” Rev. Sci. Instrum. 75(12), 5131–5135 (2004).
[CrossRef]

Kant, R.

R. Kant, “An analytical method of vector diffraction for focusing optical systems with Seidel aberrations II: Astigmatism and coma,” J. Mod. Opt. 42(2), 299–320 (1995).
[CrossRef]

R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations I. Spherical aberration, curvature of field, and distortion,” J. Mod. Opt. 40(11), 2293–2310 (1993).
[CrossRef]

Keller, J.

J. Keller, A. Schönle, and S. W. Hell, “Efficient fluorescence inhibition patterns for RESOLFT microscopy,” Opt. Express 15(6), 3361–3371 (2007).
[CrossRef] [PubMed]

K. I. Willig, J. Keller, M. Bossi, and S. W. Hell, “STED microscopy resolves nanoparticle assemblies,” N. J. Phys. 8(6), 106 (2006).
[CrossRef]

Klar, T. A.

T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(6), 066613 (2001).
[CrossRef] [PubMed]

Li, R. X.

Liu, L.

Muls, B.

Munro, P. R. T.

Neil, M. A. A.

B. R. Boruah and M. A. A. Neil, “Far field computation of an arbitrarily polarized beam using fast Fourier thrnsforms,” Opt. Commun. 282(24), 4660–4667 (2009).
[CrossRef]

B. R. Boruah and M. A. A. Neil, “Susceptibility to and correction of azimuthal aberrations in singular light beams,” Opt. Express 14(22), 10377–10385 (2006).
[CrossRef] [PubMed]

Richards, B.

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

Roichman, Y.

Schönle, A.

Senthilkumaran, P.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary spherical aberration on high-numericalaperture focusing of a Laguerre-Gaussian beam,” J. Opt. Soc. Am. A 25(6), 1307–1318 (2008).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of a vortex carrying beam with Gaussian background by an apertured system in presence of coma,” Opt. Commun. 281(5), 923–934 (2008).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of linearly-, and circularly polarized Gaussian background vortex beams by a high numerical aperture system afflicted with third-order astigmatism,” Opt. Commun. 281(24), 5939–5948 (2008).
[CrossRef]

Singh, K.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of linearly-, and circularly polarized Gaussian background vortex beams by a high numerical aperture system afflicted with third-order astigmatism,” Opt. Commun. 281(24), 5939–5948 (2008).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of a vortex carrying beam with Gaussian background by an apertured system in presence of coma,” Opt. Commun. 281(5), 923–934 (2008).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary spherical aberration on high-numericalaperture focusing of a Laguerre-Gaussian beam,” J. Opt. Soc. Am. A 25(6), 1307–1318 (2008).
[CrossRef]

Singh, R. K.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of a vortex carrying beam with Gaussian background by an apertured system in presence of coma,” Opt. Commun. 281(5), 923–934 (2008).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary spherical aberration on high-numericalaperture focusing of a Laguerre-Gaussian beam,” J. Opt. Soc. Am. A 25(6), 1307–1318 (2008).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of linearly-, and circularly polarized Gaussian background vortex beams by a high numerical aperture system afflicted with third-order astigmatism,” Opt. Commun. 281(24), 5939–5948 (2008).
[CrossRef]

Török, P.

Toyama, N.

T. Watanabe, M. Fujii, Y. Watanabe, N. Toyama, and Y. Iketaki, “Generation of a doughnut-shaped beam using a spiral phase plate,” Rev. Sci. Instrum. 75(12), 5131–5135 (2004).
[CrossRef]

Waldron, A.

Watanabe, T.

N. Bokor, Y. Iketaki, T. Watanabe, K. Daigoku, N. Davidson, and M. Fujii, “On polarization effects in fluorescence depletion microscopy,” Opt. Commun. 272(1), 263–268 (2007).
[CrossRef]

T. Watanabe, M. Fujii, Y. Watanabe, N. Toyama, and Y. Iketaki, “Generation of a doughnut-shaped beam using a spiral phase plate,” Rev. Sci. Instrum. 75(12), 5131–5135 (2004).
[CrossRef]

Watanabe, Y.

T. Watanabe, M. Fujii, Y. Watanabe, N. Toyama, and Y. Iketaki, “Generation of a doughnut-shaped beam using a spiral phase plate,” Rev. Sci. Instrum. 75(12), 5131–5135 (2004).
[CrossRef]

Westphal, V.

V. Westphal and S. W. Hell, “Nanoscale resolution in the focal plane of an optical microscope,” Phys. Rev. Lett. 94(14), 143903 (2005).
[CrossRef] [PubMed]

Wichmann, J.

Willig, K. I.

K. I. Willig, J. Keller, M. Bossi, and S. W. Hell, “STED microscopy resolves nanoparticle assemblies,” N. J. Phys. 8(6), 106 (2006).
[CrossRef]

Wolf, E.

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

Xu, Z. Z.

Appl. Opt.

J. Mod. Opt.

R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations I. Spherical aberration, curvature of field, and distortion,” J. Mod. Opt. 40(11), 2293–2310 (1993).
[CrossRef]

R. Kant, “An analytical method of vector diffraction for focusing optical systems with Seidel aberrations II: Astigmatism and coma,” J. Mod. Opt. 42(2), 299–320 (1995).
[CrossRef]

J. Opt. Soc. Am. A

N. J. Phys.

K. I. Willig, J. Keller, M. Bossi, and S. W. Hell, “STED microscopy resolves nanoparticle assemblies,” N. J. Phys. 8(6), 106 (2006).
[CrossRef]

Nat. Methods

S. W. Hell, “Microscopy and its focal switch,” Nat. Methods 6(1), 24–32 (2009).
[CrossRef] [PubMed]

Opt. Commun.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of a vortex carrying beam with Gaussian background by an apertured system in presence of coma,” Opt. Commun. 281(5), 923–934 (2008).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Focusing of linearly-, and circularly polarized Gaussian background vortex beams by a high numerical aperture system afflicted with third-order astigmatism,” Opt. Commun. 281(24), 5939–5948 (2008).
[CrossRef]

B. R. Boruah and M. A. A. Neil, “Far field computation of an arbitrarily polarized beam using fast Fourier thrnsforms,” Opt. Commun. 282(24), 4660–4667 (2009).
[CrossRef]

N. Bokor, Y. Iketaki, T. Watanabe, K. Daigoku, N. Davidson, and M. Fujii, “On polarization effects in fluorescence depletion microscopy,” Opt. Commun. 272(1), 263–268 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(6), 066613 (2001).
[CrossRef] [PubMed]

Phys. Rev. Lett.

V. Westphal and S. W. Hell, “Nanoscale resolution in the focal plane of an optical microscope,” Phys. Rev. Lett. 94(14), 143903 (2005).
[CrossRef] [PubMed]

Proc. R. Soc. Lond. A Math. Phys. Sci.

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

Rev. Sci. Instrum.

T. Watanabe, M. Fujii, Y. Watanabe, N. Toyama, and Y. Iketaki, “Generation of a doughnut-shaped beam using a spiral phase plate,” Rev. Sci. Instrum. 75(12), 5131–5135 (2004).
[CrossRef]

Science

S. W. Hell, “Far-Field Optical Nanoscopy,” Science 316(5828), 1153–1158 (2007).
[CrossRef] [PubMed]

Other

M. Born, and E. Wolf, Principles of Optics, (Cambridge University Press, Cambridge, UK, 1999).

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Figures (9)

Fig. 1
Fig. 1

Calculated intensity distribution on the focal plane and through the focus of a LC polarized Gaussian beam inserted with the vortex mask for different spherical aberration: (a) without aberration, (b) A s = 0.5 , (c) A s = 2 .

Fig. 2
Fig. 2

Calculated intensity distribution on the focal plane and through the focus of a linearly polarized Gaussian beam inserted with the semi-circular λ / 2 phase mask for different spherical aberration: (a) without aberration, (b) A s = 0.5 , (c) A s = 2 .

Fig. 3
Fig. 3

Calculated intensity distribution on the x z plane of a LC polarized Gaussian beam inserted with the central λ / 2 phase mask for different spherical aberration: (a) without aberration, (b) A s = 0.5 , (c) A s = 2 . Figure (d) is normalized intensity profiles through central zero intensity of this pattern for different spherical aberrations.

Fig. 4
Fig. 4

Calculated intensity distribution on the focal plane and through the focus of a LC polarized Gaussian beam inserted with the vortex mask for different coma: (a) A c = 0.15 , (b) A c = 0.5 , (c) A c = 1 .

Fig. 5
Fig. 5

Calculated intensity distribution through the focus and on the focal plane of a LC polarized Gaussian beam inserted with the central λ / 2 phase mask for different coma: (a) A c = 0.15 , (b) A c = 0.5 , (c) A c = 1 .

Fig. 6
Fig. 6

Calculated intensity distribution on the focal plane and through the focus of a linearly polarized Gaussian beam inserted with the semi-circular λ / 2 phase mask for different coma: (a) A c = 0.15 , (b) A c = 0.5 , (c) A c = 1 . Figure (d) is normalized intensity profiles through central zero intensity of this pattern for different coma.

Fig. 7
Fig. 7

Calculated intensity distribution on the focal plane and through the focus of a LC polarized Gaussian beam inserted with the vortex mask for different astigmatism: (a) A a = 0.15 , (b) A a = 0.5 , (c) A a = 1 .

Fig. 8
Fig. 8

Calculated intensity distribution on the x z plane of a LC polarized Gaussian beam inserted with the central λ / 2 phase mask for different astigmatism: (a) A a = 0.15 , (b) A a = 0.5 , (c) A a = 1 .

Fig. 9
Fig. 9

Calculated intensity distribution on the focal plane and through the focus of a linearly polarized Gaussian beam inserted with the semi-circular λ / 2 phase mask for different astigmatism: (a) A a = 0.15 , (b) A a = 0.5 , (c) A a = 1 . Figure (d) is normalized intensity profiles through central zero intensity of this pattern for different astigmatism.

Equations (7)

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

E 0 ( γ , θ ) = A 0 exp ( γ 2 sin 2 θ sin 2 α )
E ( p ) = ( E x E y E z ) = i f λ 0 α 0 2 π E 0 cos θ A 1 ( θ , ϕ ) exp [ i k ( x sin θ cos ϕ + y sin θ sin ϕ + z cos θ ) ] × φ s ( θ , ϕ ) [ cos θ cos 2 ϕ + sin 2 ϕ cos ϕ sin ϕ ( cos θ 1 ) sin θ cos θ ] sin θ d θ d ϕ
A 1 ( θ , ϕ ) = exp [ i k A s ( sin θ sin α ) 4 ]
A 1 ( θ , ϕ ) = exp [ i k A c ( sin θ sin α ) 3 cos ϕ ]
A 1 ( θ , ϕ ) = exp [ i k A a ( sin θ sin α ) 2 cos 2 ϕ ]
E ( p ) = ( E x E y E z ) = i f λ 0 α 0 2 π E 0 cos θ A 1 ( θ , ϕ ) exp [ i k ( x sin θ cos ϕ + y sin θ sin ϕ + z cos θ ) ] × φ s ( θ , ϕ ) [ cos θ cos 2 ϕ + sin 2 ϕ + i sin ϕ cos ϕ ( cos θ 1 ) cos ϕ sin ϕ ( cos θ 1 ) + i ( cos θ sin 2 ϕ + cos 2 ϕ ) sin θ ( cos θ + i sin ϕ ) ] sin θ d θ d ϕ
d λ / ( 2 N A 1 + I S T E D max / I s )

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