Abstract

Recently proposed subtraction microscopy provides a simple and compact approach to enhance the spatial resolution. However, inevitable imperfect overlap of the Gaussian and donut spots could deteriorate the quality of the subtracted images, while this effect is rarely investigated quantitatively. Based on vector diffraction theory, the effects of spatial displacements on the image resolution, signal intensity, and subtraction threshold are studied with several microscope parameters here. The peak-position shift is found to be a useful and convenient parameter to inspect and calibrate the spatial overlap of excitation spots in such microscopes.

© 2015 Optical Society of America

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References

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2015 (1)

N. Wang, J. Miyazaki, J. He, K. Seto, and T. Kobayashi, “Sub-diffraction-limit imaging using mode multiplexing,” Opt. Commun. 343, 28–33 (2015).
[Crossref]

2014 (7)

2013 (2)

C. F. Kuang, S. Li, W. Liu, X. Hao, Z. T. Gu, Y. F. Wang, J. H. Ge, H. F. Li, and X. Liu, “Breaking the diffraction barrier using fluorescence emission difference microscopy,” Sci. Rep. 3, 1441 (2013).
[Crossref]

H. Dehez, M. Piche, and Y. D. Koninck, “Resolution and contrast enhancement in laser scanning microscopy using dark beam imaging,” Opt. Express 21(13), 15912–15925 (2013).
[Crossref] [PubMed]

2009 (3)

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” P. Natl. Acad. Sci. USA 106(52), 22287–22292 (2009).
[Crossref]

E. Rittweger, K. Y. Han, S. E. Irvine, C. Eggeling, and S. W. Hell, “STED microscopy reveals crystal colour centres with nanometric resolution,” Nat. Photonics 3, 144–147 (2009).
[Crossref]

S. Deng, L. Liu, Y. Cheng, R. Li, and 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 (1)

2006 (3)

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. L. 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. W. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Meth. 3, 793–796 (2006).
[Crossref]

A. Sharonov and R. M. Hochstrasser, “Wide-field sub-diffraction imaging by accumulated binding of diffusing probes,” P. Natl. Acad. Sci. USA 103(50), 18911–18916 (2006).
[Crossref]

2004 (1)

M. P. Gordon, T. Ha, and P. R. Selvin, “Single-molecule high-resolution imaging with photobleaching,” P. Natl. Acad. Sci. USA 101(17), 6462–6465 (2004).
[Crossref]

2003 (2)

2000 (1)

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2000).
[Crossref] [PubMed]

1994 (1)

1959 (1)

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 253, 358–379 (1959).
[Crossref]

Bates, M.

M. J. Rust, M. Bates, and X. W. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Meth. 3, 793–796 (2006).
[Crossref]

Betzig, E.

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

Bonifacino, J. S.

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

Cheng, Y.

Colyer, R.

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” P. Natl. Acad. Sci. USA 106(52), 22287–22292 (2009).
[Crossref]

Davidson, M. W.

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

Dehez, H.

Deng, S.

Dertinger, T.

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” P. Natl. Acad. Sci. USA 106(52), 22287–22292 (2009).
[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]

Eggeling, C.

E. Rittweger, K. Y. Han, S. E. Irvine, C. Eggeling, and S. W. Hell, “STED microscopy reveals crystal colour centres with nanometric resolution,” Nat. Photonics 3, 144–147 (2009).
[Crossref]

Enderlein, J.

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” P. Natl. Acad. Sci. USA 106(52), 22287–22292 (2009).
[Crossref]

Gan, X.

Ganic, D.

Ge, J. H.

C. F. Kuang, S. Li, W. Liu, X. Hao, Z. T. Gu, Y. F. Wang, J. H. Ge, H. F. Li, and X. Liu, “Breaking the diffraction barrier using fluorescence emission difference microscopy,” Sci. Rep. 3, 1441 (2013).
[Crossref]

Gordon, M. P.

M. P. Gordon, T. Ha, and P. R. Selvin, “Single-molecule high-resolution imaging with photobleaching,” P. Natl. Acad. Sci. USA 101(17), 6462–6465 (2004).
[Crossref]

Gu, M.

Gu, Z. T.

C. F. Kuang, S. Li, W. Liu, X. Hao, Z. T. Gu, Y. F. Wang, J. H. Ge, H. F. Li, and X. Liu, “Breaking the diffraction barrier using fluorescence emission difference microscopy,” Sci. Rep. 3, 1441 (2013).
[Crossref]

Gustafsson, M. G. L.

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2000).
[Crossref] [PubMed]

Ha, T.

M. P. Gordon, T. Ha, and P. R. Selvin, “Single-molecule high-resolution imaging with photobleaching,” P. Natl. Acad. Sci. USA 101(17), 6462–6465 (2004).
[Crossref]

Han, K. Y.

E. Rittweger, K. Y. Han, S. E. Irvine, C. Eggeling, and S. W. Hell, “STED microscopy reveals crystal colour centres with nanometric resolution,” Nat. Photonics 3, 144–147 (2009).
[Crossref]

Hao, X.

C. F. Kuang, S. Li, W. Liu, X. Hao, Z. T. Gu, Y. F. Wang, J. H. Ge, H. F. Li, and X. Liu, “Breaking the diffraction barrier using fluorescence emission difference microscopy,” Sci. Rep. 3, 1441 (2013).
[Crossref]

Hayashi-Takagi, A.

He, J.

N. Wang, J. Miyazaki, J. He, K. Seto, and T. Kobayashi, “Sub-diffraction-limit imaging using mode multiplexing,” Opt. Commun. 343, 28–33 (2015).
[Crossref]

Hell, S. W.

E. Rittweger, K. Y. Han, S. E. Irvine, C. Eggeling, and S. W. Hell, “STED microscopy reveals crystal colour centres with nanometric resolution,” Nat. Photonics 3, 144–147 (2009).
[Crossref]

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]

Hess, H. F.

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

Hochstrasser, R. M.

A. Sharonov and R. M. Hochstrasser, “Wide-field sub-diffraction imaging by accumulated binding of diffusing probes,” P. Natl. Acad. Sci. USA 103(50), 18911–18916 (2006).
[Crossref]

Irvine, S. E.

E. Rittweger, K. Y. Han, S. E. Irvine, C. Eggeling, and S. W. Hell, “STED microscopy reveals crystal colour centres with nanometric resolution,” Nat. Photonics 3, 144–147 (2009).
[Crossref]

Iyer, G.

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” P. Natl. Acad. Sci. USA 106(52), 22287–22292 (2009).
[Crossref]

Kasai, H.

Kawasumi, K.

Kobayashi, T.

Koninck, Y. D.

Kozawa, Y.

Kuang, C.

Kuang, C. F.

C. F. Kuang, S. Li, W. Liu, X. Hao, Z. T. Gu, Y. F. Wang, J. H. Ge, H. F. Li, and X. Liu, “Breaking the diffraction barrier using fluorescence emission difference microscopy,” Sci. Rep. 3, 1441 (2013).
[Crossref]

Lerman, G. M.

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]

Levy, U.

Li, C.

J. Yao, L. Wang, C. Li, C. Zhang, and L. V. Wang, “Photoimprint photoacoustic microscopy for three-dimensional label-free subdiffraction imaging,” Phys. Rev. Lett. 112, 014302 (2014).
[Crossref] [PubMed]

Li, H. F.

C. F. Kuang, S. Li, W. Liu, X. Hao, Z. T. Gu, Y. F. Wang, J. H. Ge, H. F. Li, and X. Liu, “Breaking the diffraction barrier using fluorescence emission difference microscopy,” Sci. Rep. 3, 1441 (2013).
[Crossref]

Li, R.

Li, S.

C. F. Kuang, S. Li, W. Liu, X. Hao, Z. T. Gu, Y. F. Wang, J. H. Ge, H. F. Li, and X. Liu, “Breaking the diffraction barrier using fluorescence emission difference microscopy,” Sci. Rep. 3, 1441 (2013).
[Crossref]

Lindwasser, O. W.

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

Liu, L.

Liu, W.

C. F. Kuang, S. Li, W. Liu, X. Hao, Z. T. Gu, Y. F. Wang, J. H. Ge, H. F. Li, and X. Liu, “Breaking the diffraction barrier using fluorescence emission difference microscopy,” Sci. Rep. 3, 1441 (2013).
[Crossref]

Liu, X.

S. You, C. Kuang, Z. Rong, and X. Liu, “Eliminating deformations in fluorescence emission difference microscopy,” Opt. Express 22(21), 26375–26385 (2014).
[Crossref] [PubMed]

C. F. Kuang, S. Li, W. Liu, X. Hao, Z. T. Gu, Y. F. Wang, J. H. Ge, H. F. Li, and X. Liu, “Breaking the diffraction barrier using fluorescence emission difference microscopy,” Sci. Rep. 3, 1441 (2013).
[Crossref]

Miyazaki, J.

Olenych, S.

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

Patterson, G. H.

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

Piche, M.

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]

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 253, 358–379 (1959).
[Crossref]

Rittweger, E.

E. Rittweger, K. Y. Han, S. E. Irvine, C. Eggeling, and S. W. Hell, “STED microscopy reveals crystal colour centres with nanometric resolution,” Nat. Photonics 3, 144–147 (2009).
[Crossref]

Rong, Z.

Rust, M. J.

M. J. Rust, M. Bates, and X. W. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Meth. 3, 793–796 (2006).
[Crossref]

Sato, S.

Schwartz, J. L.

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

Segawa, S.

Selvin, P. R.

M. P. Gordon, T. Ha, and P. R. Selvin, “Single-molecule high-resolution imaging with photobleaching,” P. Natl. Acad. Sci. USA 101(17), 6462–6465 (2004).
[Crossref]

Seto, K.

N. Wang, J. Miyazaki, J. He, K. Seto, and T. Kobayashi, “Sub-diffraction-limit imaging using mode multiplexing,” Opt. Commun. 343, 28–33 (2015).
[Crossref]

Sharonov, A.

A. Sharonov and R. M. Hochstrasser, “Wide-field sub-diffraction imaging by accumulated binding of diffusing probes,” P. Natl. Acad. Sci. USA 103(50), 18911–18916 (2006).
[Crossref]

Sougrat, R.

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

Tsurui, H.

Wang, L.

J. Yao, L. Wang, C. Li, C. Zhang, and L. V. Wang, “Photoimprint photoacoustic microscopy for three-dimensional label-free subdiffraction imaging,” Phys. Rev. Lett. 112, 014302 (2014).
[Crossref] [PubMed]

Wang, L. V.

J. Yao, L. Wang, C. Li, C. Zhang, and L. V. Wang, “Photoimprint photoacoustic microscopy for three-dimensional label-free subdiffraction imaging,” Phys. Rev. Lett. 112, 014302 (2014).
[Crossref] [PubMed]

Wang, N.

N. Wang, J. Miyazaki, J. He, K. Seto, and T. Kobayashi, “Sub-diffraction-limit imaging using mode multiplexing,” Opt. Commun. 343, 28–33 (2015).
[Crossref]

N. Wang and T. Kobayashi, “Numerical study of the subtraction threshold for fluorescence difference microscopy,” Opt. Express 22(23), 28819–28830 (2014).
[Crossref] [PubMed]

Wang, Y. F.

C. F. Kuang, S. Li, W. Liu, X. Hao, Z. T. Gu, Y. F. Wang, J. H. Ge, H. F. Li, and X. Liu, “Breaking the diffraction barrier using fluorescence emission difference microscopy,” Sci. Rep. 3, 1441 (2013).
[Crossref]

Weiss, S.

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” P. Natl. Acad. Sci. USA 106(52), 22287–22292 (2009).
[Crossref]

Wichmann, J.

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 253, 358–379 (1959).
[Crossref]

Xu, Z.

Yao, J.

J. Yao, L. Wang, C. Li, C. Zhang, and L. V. Wang, “Photoimprint photoacoustic microscopy for three-dimensional label-free subdiffraction imaging,” Phys. Rev. Lett. 112, 014302 (2014).
[Crossref] [PubMed]

You, S.

Zhang, C.

J. Yao, L. Wang, C. Li, C. Zhang, and L. V. Wang, “Photoimprint photoacoustic microscopy for three-dimensional label-free subdiffraction imaging,” Phys. Rev. Lett. 112, 014302 (2014).
[Crossref] [PubMed]

Zhuang, X. W.

M. J. Rust, M. Bates, and X. W. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Meth. 3, 793–796 (2006).
[Crossref]

J. Microsc. (1)

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2000).
[Crossref] [PubMed]

Nat. Meth. (1)

M. J. Rust, M. Bates, and X. W. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Meth. 3, 793–796 (2006).
[Crossref]

Nat. Photonics (1)

E. Rittweger, K. Y. Han, S. E. Irvine, C. Eggeling, and S. W. Hell, “STED microscopy reveals crystal colour centres with nanometric resolution,” Nat. Photonics 3, 144–147 (2009).
[Crossref]

Opt. Commun. (1)

N. Wang, J. Miyazaki, J. He, K. Seto, and T. Kobayashi, “Sub-diffraction-limit imaging using mode multiplexing,” Opt. Commun. 343, 28–33 (2015).
[Crossref]

Opt. Express (7)

Opt. Lett. (4)

P. Natl. Acad. Sci. USA (3)

M. P. Gordon, T. Ha, and P. R. Selvin, “Single-molecule high-resolution imaging with photobleaching,” P. Natl. Acad. Sci. USA 101(17), 6462–6465 (2004).
[Crossref]

A. Sharonov and R. M. Hochstrasser, “Wide-field sub-diffraction imaging by accumulated binding of diffusing probes,” P. Natl. Acad. Sci. USA 103(50), 18911–18916 (2006).
[Crossref]

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” P. Natl. Acad. Sci. USA 106(52), 22287–22292 (2009).
[Crossref]

Phys. Rev. Lett. (2)

J. Yao, L. Wang, C. Li, C. Zhang, and L. V. Wang, “Photoimprint photoacoustic microscopy for three-dimensional label-free subdiffraction imaging,” Phys. Rev. Lett. 112, 014302 (2014).
[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]

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

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 253, 358–379 (1959).
[Crossref]

Sci. Rep. (1)

C. F. Kuang, S. Li, W. Liu, X. Hao, Z. T. Gu, Y. F. Wang, J. H. Ge, H. F. Li, and X. Liu, “Breaking the diffraction barrier using fluorescence emission difference microscopy,” Sci. Rep. 3, 1441 (2013).
[Crossref]

Science (1)

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

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Figures (7)

Fig. 1
Fig. 1 Schematics of spatial displacements in subtraction microscope. (a)–(d) Excitation spots with lateral misalignment of δx, subtracted images with factor of 0.3 while misaligned values are δx1=0.05λ and δx2=0.2λ, and the cross sectioning profiles. (e)–(f) Excitation spots with axial misalignment of δz, subtracted images with factor of 0.3 while misaligned values are δz1=0.05λ and δz2=0.2λ, and their cross sectioning profiles.
Fig. 2
Fig. 2 (a) Cross sectioning lines of lateral shifted excitation spots. (b) Subtracted results with γ=0, 0.3, 1. (c)–(f) Evolution of subtracted FWHM values, normalized peak intensities, maximum values of negative sidebands and the subtracted peak position shifts versus the subtraction factors with excitation displacements of δx=0, 0.05λ, 0.1λ, 0.2λ. (g)–(h) Subtraction threshold lines and the threshold peak intensity factors with excitation displacements of δx=0, 0.05λ, 0.1λ, 0.2λ. (i) schematic of key parameters in the subtracted microscopes. The preset sample size is r=0.1λ.
Fig. 3
Fig. 3 (a) Intensities along the axial shifted excitation spots. (b) Subtraction results of them with γ=0, 0.3, 1. (c)–(e) Evolution of subtracted FWHM values, normalized peak intensities and maximum values of negative signal versus the subtraction factors with excitation displacements of δz=0, 0.2λ, 0.3λ, 0.4λ. (f)–(g) Subtraction threshold lines and the peak intensity factors with excitation displacements. (h)–(i) Subtraction threshold lines and the peak intensity factors with only IG or ID shifted from central focal plane. The preset sample size is r=0.1λ.
Fig. 4
Fig. 4 Simulation results with preset sample size of r=0.3λ. (a)–(d) FWHM values, normalized peak intensities, normalized absolute values of maximum negative signals and peak position shifts versus subtraction factors with lateral displacements of δx=0, 0.05λ, 0.1λ, 0.2λ. (e)–(g) FWHM values, normalized peak intensities and normalized absolute values of maximum negative signals versus subtraction factors with axial displacements of δz=0, 0.2λ, 0.3λ, 0.4λ. (h)–(k) Corresponding subtraction threshold lines and peak intensity factors with lateral and axial displacements.
Fig. 5
Fig. 5 Results of simulation with objective NA=0.9. (a)–(d) FWHM, peak intensities, Maximum negative values and peak position shifts versus the subtraction factors with lateral displacements. (e)–(g) FWHM, peak intensities, maximum negative value versus the subtraction factors with axial displacements. (h)–(k) Subtraction threshold lines and peak intensity factors with lateral and axial displacements, respectively. The preset sample size is r=0.4λ.
Fig. 6
Fig. 6 Simulation results with radial and azimuthal polarizations. (a)–(d) FWHM, peak intensities, Maximum negative values and peak position shifts versus the subtraction factors with lateral displacements. (e)–(g) FWHM, peak intensities, maximum negative value versus the subtraction factors with axial displacements. (h)–(k) Subtraction threshold lines and peak intensity factors with lateral and axial displacements, respectively. Sample size is r=0.1λ.
Fig. 7
Fig. 7 Simulations with 2D position shifts with beads of FWHM=200, 150, 100 and 50 nm (from left to right, top to bottom in the subtracted images). (a)–(c) Evolution of peak position shifts in X, Y direction and the peak intensity values with IG displaced with δx=12 nm and δy=20 nm. Similarly, (d)–(e) Results with δx=40 nm and δy=−24 nm. (g)–(i) Results with δx=−52 nm and δy=−56 nm. (j)–(l) Results without misalignment.

Tables (1)

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Table 1 Set values of IG and ID.

Equations (10)

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E x , y , z ( x , y , z ) = i C λ 0 2 π 0 arcsin ( NA / n ) A AMP A Phase A L ( P x P y P z ) sin θ e i k { z cos θ + x 2 + y 2 sin θ cos [ φ arctan ( y / x ) ] } d θ d φ
I G , D ( x , y , 0 ) = | E x | 2 + | E y | 2 + | E z | 2
A L = cos θ ( 1 + ( cos θ 1 ) cos 2 φ ( cos θ 1 ) cos φ sin φ sin θ cos φ ( cos θ 1 ) cos φ sin φ 1 + ( cos θ 1 ) sin 2 φ sin θ sin φ sin θ cos φ sin θ sin φ cos θ )
O b ( x , y , r ) = C 0 e ( 4 ln 2 ) x 2 + y 2 r 2
I RAW ( x , y , z , δ x , δ z , r , γ ) = I G ( x + δ x / 2 , y , z + δ z / 2 ) * Ob ( x , y , r ) γ I D ( x δ x / 2 , y , z δ z / 2 ) * Ob ( x , y , r ) I ( x , y , z , δ x , δ z , r , γ ) = { I RAW ( x , y , z , δ x , δ z , r , γ ) I RAW ( x , y , z , δ x , δ z , r , γ ) 0 0 I RAW ( x , y , z , δ x , δ z , r , γ ) < 0
I peak = MAX [ I ( x , y , z , δ x , δ z , r , γ ) ] MAX [ I ( x , y , z , δ x , δ z , r , 0 ) ]
| I Neg | = { 0 MIN [ I RAW ( x , y , z , δ x , δ z , r , γ ) ] 0 | MIN [ I RAW ( x , y , z , δ x , δ z , r , γ ) ] | MAX [ I ( x , y , z , δ x , δ z , r , γ ) ] MIN [ I RAW ( x , y , z , δ x , δ z , r , γ ) ] < 0
FWHM Ob ( x , y , r 0 ) = FWHM I ( x , y , z , δ x , δ z , r 0 , γ 0 )
I P = MAX [ I ( x , y , z , δ x , δ z , r 0 , γ 0 ) ] MAX [ I ( x , y , z , δ x , δ z , r 0 , 0 ) ]
( Δ x , Δ y ) = ( X , Y ) MAX [ I ( x , y , z , δ x , δ y , r , γ ) ] ( X 0 , Y 0 ) MAX [ I ( x , y , z , δ x , δ y , r , 0 ) ]

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