Abstract

We demonstrate an all-optical plasmonic structured illumination microscopy (PSIM) technique. A set of plasmonic standing-wave patterns is excited by amplitude-modified optical vortices (OVs), which have fractional topological charges for precise phase shift of {-2π/3, 0, 2π/3}. A specially designed optical aperture is introduced to modify the OVs in order to improve the uniformity of interference patterns. The imaging results of fluorescent beads reveal a sub-100nm resolving capability in aqueous environment. This PSIM technique as a structure-free, wide-field and super-resolved imaging technique is of great potential for low-cost biological dynamic imaging applications.

© 2015 Optical Society of America

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References

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  1. C. Cremer and T. Cremer, “Considerations on a laser-scanning-microscope with high resolution and depth of field,” Microsc. Acta 81(1), 31–44 (1978).
    [PubMed]
  2. C. J. R. Sheppard and T. Wilson, “The theory of the direct-view confocal microscope,” J. Microsc. 124(2), 107–117 (1981).
    [Crossref] [PubMed]
  3. E. A. Ash and G. Nicholls, “Super-resolution aperture scanning microscope,” Nature 237(5357), 510–512 (1972).
    [Crossref] [PubMed]
  4. A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, “Development of a 500 Å spatial resolution light microscope,” Ultramicroscopy 13(3), 227–231 (1984).
    [Crossref]
  5. 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]
  6. V. Westphal, S. O. Rizzoli, M. A. Lauterbach, D. Kamin, R. Jahn, and S. W. Hell, “Video-rate far-field optical nanoscopy dissects synaptic vesicle movement,” Science 320(5873), 246–249 (2008).
    [Crossref] [PubMed]
  7. R. Heintzmann and C. G. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE 3568, 185–196 (1999).
    [Crossref]
  8. M. G. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(2), 82–87 (2000).
    [Crossref] [PubMed]
  9. M. G. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. U.S.A. 102(37), 13081–13086 (2005).
    [Crossref] [PubMed]
  10. E. Chung, D. Kim, Y. Cui, Y. H. Kim, and P. T. So, “Two-dimensional standing wave total internal reflection fluorescence microscopy: superresolution imaging of single molecular and biological specimens,” Biophys. J. 93(5), 1747–1757 (2007).
    [Crossref] [PubMed]
  11. E. Chung, Y. H. Kim, W. T. Tang, C. J. Sheppard, and P. T. So, “Wide-field extended-resolution fluorescence microscopy with standing surface-plasmon-resonance waves,” Opt. Lett. 34(15), 2366–2368 (2009).
    [Crossref] [PubMed]
  12. F. Wei, D. Lu, H. Shen, W. Wan, J. L. Ponsetto, E. Huang, and Z. Liu, “Wide field super-resolution surface imaging through plasmonic structured illumination microscopy,” Nano Lett. 14(8), 4634–4639 (2014).
    [Crossref] [PubMed]
  13. J. L. Ponsetto, F. Wei, and Z. Liu, “Localized plasmon assisted structured illumination microscopy for wide-field high-speed dispersion-independent super resolution imaging,” Nanoscale 6(11), 5807–5812 (2014).
    [Crossref] [PubMed]
  14. P. S. Tan, X. C. Yuan, G. H. Yuan, and Q. Wang, “High-resolution wide-field standing-wave surface plasmon resonance fluorescence microscopy with optical vortices,” Appl. Phys. Lett. 97(24), 241109 (2010).
    [Crossref]
  15. M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6(2), 259–268 (2004).
    [Crossref]
  16. N. Calander, “Theory and simulation of surface plasmon-coupled directional emission from fluorophores at planar structures,” Anal. Chem. 76(8), 2168–2173 (2004).
    [Crossref] [PubMed]
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2014 (2)

F. Wei, D. Lu, H. Shen, W. Wan, J. L. Ponsetto, E. Huang, and Z. Liu, “Wide field super-resolution surface imaging through plasmonic structured illumination microscopy,” Nano Lett. 14(8), 4634–4639 (2014).
[Crossref] [PubMed]

J. L. Ponsetto, F. Wei, and Z. Liu, “Localized plasmon assisted structured illumination microscopy for wide-field high-speed dispersion-independent super resolution imaging,” Nanoscale 6(11), 5807–5812 (2014).
[Crossref] [PubMed]

2010 (1)

P. S. Tan, X. C. Yuan, G. H. Yuan, and Q. Wang, “High-resolution wide-field standing-wave surface plasmon resonance fluorescence microscopy with optical vortices,” Appl. Phys. Lett. 97(24), 241109 (2010).
[Crossref]

2009 (1)

2008 (1)

V. Westphal, S. O. Rizzoli, M. A. Lauterbach, D. Kamin, R. Jahn, and S. W. Hell, “Video-rate far-field optical nanoscopy dissects synaptic vesicle movement,” Science 320(5873), 246–249 (2008).
[Crossref] [PubMed]

2007 (1)

E. Chung, D. Kim, Y. Cui, Y. H. Kim, and P. T. So, “Two-dimensional standing wave total internal reflection fluorescence microscopy: superresolution imaging of single molecular and biological specimens,” Biophys. J. 93(5), 1747–1757 (2007).
[Crossref] [PubMed]

2005 (1)

M. G. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. U.S.A. 102(37), 13081–13086 (2005).
[Crossref] [PubMed]

2004 (2)

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6(2), 259–268 (2004).
[Crossref]

N. Calander, “Theory and simulation of surface plasmon-coupled directional emission from fluorophores at planar structures,” Anal. Chem. 76(8), 2168–2173 (2004).
[Crossref] [PubMed]

2001 (1)

2000 (1)

M. G. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(2), 82–87 (2000).
[Crossref] [PubMed]

1999 (1)

R. Heintzmann and C. G. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE 3568, 185–196 (1999).
[Crossref]

1994 (1)

1984 (1)

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, “Development of a 500 Å spatial resolution light microscope,” Ultramicroscopy 13(3), 227–231 (1984).
[Crossref]

1981 (1)

C. J. R. Sheppard and T. Wilson, “The theory of the direct-view confocal microscope,” J. Microsc. 124(2), 107–117 (1981).
[Crossref] [PubMed]

1978 (1)

C. Cremer and T. Cremer, “Considerations on a laser-scanning-microscope with high resolution and depth of field,” Microsc. Acta 81(1), 31–44 (1978).
[PubMed]

1972 (1)

E. A. Ash and G. Nicholls, “Super-resolution aperture scanning microscope,” Nature 237(5357), 510–512 (1972).
[Crossref] [PubMed]

Ash, E. A.

E. A. Ash and G. Nicholls, “Super-resolution aperture scanning microscope,” Nature 237(5357), 510–512 (1972).
[Crossref] [PubMed]

Berry, M. V.

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6(2), 259–268 (2004).
[Crossref]

Calander, N.

N. Calander, “Theory and simulation of surface plasmon-coupled directional emission from fluorophores at planar structures,” Anal. Chem. 76(8), 2168–2173 (2004).
[Crossref] [PubMed]

Chung, E.

E. Chung, Y. H. Kim, W. T. Tang, C. J. Sheppard, and P. T. So, “Wide-field extended-resolution fluorescence microscopy with standing surface-plasmon-resonance waves,” Opt. Lett. 34(15), 2366–2368 (2009).
[Crossref] [PubMed]

E. Chung, D. Kim, Y. Cui, Y. H. Kim, and P. T. So, “Two-dimensional standing wave total internal reflection fluorescence microscopy: superresolution imaging of single molecular and biological specimens,” Biophys. J. 93(5), 1747–1757 (2007).
[Crossref] [PubMed]

Cremer, C.

C. Cremer and T. Cremer, “Considerations on a laser-scanning-microscope with high resolution and depth of field,” Microsc. Acta 81(1), 31–44 (1978).
[PubMed]

Cremer, C. G.

R. Heintzmann and C. G. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE 3568, 185–196 (1999).
[Crossref]

Cremer, T.

C. Cremer and T. Cremer, “Considerations on a laser-scanning-microscope with high resolution and depth of field,” Microsc. Acta 81(1), 31–44 (1978).
[PubMed]

Cui, Y.

E. Chung, D. Kim, Y. Cui, Y. H. Kim, and P. T. So, “Two-dimensional standing wave total internal reflection fluorescence microscopy: superresolution imaging of single molecular and biological specimens,” Biophys. J. 93(5), 1747–1757 (2007).
[Crossref] [PubMed]

Dong, C. Y.

Gustafsson, M. G.

M. G. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. U.S.A. 102(37), 13081–13086 (2005).
[Crossref] [PubMed]

M. G. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(2), 82–87 (2000).
[Crossref] [PubMed]

Harootunian, A.

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, “Development of a 500 Å spatial resolution light microscope,” Ultramicroscopy 13(3), 227–231 (1984).
[Crossref]

Heintzmann, R.

R. Heintzmann and C. G. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE 3568, 185–196 (1999).
[Crossref]

Hell, S. W.

V. Westphal, S. O. Rizzoli, M. A. Lauterbach, D. Kamin, R. Jahn, and S. W. Hell, “Video-rate far-field optical nanoscopy dissects synaptic vesicle movement,” Science 320(5873), 246–249 (2008).
[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]

Huang, E.

F. Wei, D. Lu, H. Shen, W. Wan, J. L. Ponsetto, E. Huang, and Z. Liu, “Wide field super-resolution surface imaging through plasmonic structured illumination microscopy,” Nano Lett. 14(8), 4634–4639 (2014).
[Crossref] [PubMed]

Isaacson, M.

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, “Development of a 500 Å spatial resolution light microscope,” Ultramicroscopy 13(3), 227–231 (1984).
[Crossref]

Jahn, R.

V. Westphal, S. O. Rizzoli, M. A. Lauterbach, D. Kamin, R. Jahn, and S. W. Hell, “Video-rate far-field optical nanoscopy dissects synaptic vesicle movement,” Science 320(5873), 246–249 (2008).
[Crossref] [PubMed]

Kamin, D.

V. Westphal, S. O. Rizzoli, M. A. Lauterbach, D. Kamin, R. Jahn, and S. W. Hell, “Video-rate far-field optical nanoscopy dissects synaptic vesicle movement,” Science 320(5873), 246–249 (2008).
[Crossref] [PubMed]

Kim, D.

E. Chung, D. Kim, Y. Cui, Y. H. Kim, and P. T. So, “Two-dimensional standing wave total internal reflection fluorescence microscopy: superresolution imaging of single molecular and biological specimens,” Biophys. J. 93(5), 1747–1757 (2007).
[Crossref] [PubMed]

Kim, Y. H.

E. Chung, Y. H. Kim, W. T. Tang, C. J. Sheppard, and P. T. So, “Wide-field extended-resolution fluorescence microscopy with standing surface-plasmon-resonance waves,” Opt. Lett. 34(15), 2366–2368 (2009).
[Crossref] [PubMed]

E. Chung, D. Kim, Y. Cui, Y. H. Kim, and P. T. So, “Two-dimensional standing wave total internal reflection fluorescence microscopy: superresolution imaging of single molecular and biological specimens,” Biophys. J. 93(5), 1747–1757 (2007).
[Crossref] [PubMed]

Kwon, H. S.

Lauterbach, M. A.

V. Westphal, S. O. Rizzoli, M. A. Lauterbach, D. Kamin, R. Jahn, and S. W. Hell, “Video-rate far-field optical nanoscopy dissects synaptic vesicle movement,” Science 320(5873), 246–249 (2008).
[Crossref] [PubMed]

Lewis, A.

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, “Development of a 500 Å spatial resolution light microscope,” Ultramicroscopy 13(3), 227–231 (1984).
[Crossref]

Liu, Z.

F. Wei, D. Lu, H. Shen, W. Wan, J. L. Ponsetto, E. Huang, and Z. Liu, “Wide field super-resolution surface imaging through plasmonic structured illumination microscopy,” Nano Lett. 14(8), 4634–4639 (2014).
[Crossref] [PubMed]

J. L. Ponsetto, F. Wei, and Z. Liu, “Localized plasmon assisted structured illumination microscopy for wide-field high-speed dispersion-independent super resolution imaging,” Nanoscale 6(11), 5807–5812 (2014).
[Crossref] [PubMed]

Lu, D.

F. Wei, D. Lu, H. Shen, W. Wan, J. L. Ponsetto, E. Huang, and Z. Liu, “Wide field super-resolution surface imaging through plasmonic structured illumination microscopy,” Nano Lett. 14(8), 4634–4639 (2014).
[Crossref] [PubMed]

Muray, A.

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, “Development of a 500 Å spatial resolution light microscope,” Ultramicroscopy 13(3), 227–231 (1984).
[Crossref]

Nicholls, G.

E. A. Ash and G. Nicholls, “Super-resolution aperture scanning microscope,” Nature 237(5357), 510–512 (1972).
[Crossref] [PubMed]

Ponsetto, J. L.

F. Wei, D. Lu, H. Shen, W. Wan, J. L. Ponsetto, E. Huang, and Z. Liu, “Wide field super-resolution surface imaging through plasmonic structured illumination microscopy,” Nano Lett. 14(8), 4634–4639 (2014).
[Crossref] [PubMed]

J. L. Ponsetto, F. Wei, and Z. Liu, “Localized plasmon assisted structured illumination microscopy for wide-field high-speed dispersion-independent super resolution imaging,” Nanoscale 6(11), 5807–5812 (2014).
[Crossref] [PubMed]

Rizzoli, S. O.

V. Westphal, S. O. Rizzoli, M. A. Lauterbach, D. Kamin, R. Jahn, and S. W. Hell, “Video-rate far-field optical nanoscopy dissects synaptic vesicle movement,” Science 320(5873), 246–249 (2008).
[Crossref] [PubMed]

Shen, H.

F. Wei, D. Lu, H. Shen, W. Wan, J. L. Ponsetto, E. Huang, and Z. Liu, “Wide field super-resolution surface imaging through plasmonic structured illumination microscopy,” Nano Lett. 14(8), 4634–4639 (2014).
[Crossref] [PubMed]

Sheppard, C. J.

Sheppard, C. J. R.

C. J. R. Sheppard and T. Wilson, “The theory of the direct-view confocal microscope,” J. Microsc. 124(2), 107–117 (1981).
[Crossref] [PubMed]

So, P. T.

Tan, P. S.

P. S. Tan, X. C. Yuan, G. H. Yuan, and Q. Wang, “High-resolution wide-field standing-wave surface plasmon resonance fluorescence microscopy with optical vortices,” Appl. Phys. Lett. 97(24), 241109 (2010).
[Crossref]

Tang, W. T.

Wan, W.

F. Wei, D. Lu, H. Shen, W. Wan, J. L. Ponsetto, E. Huang, and Z. Liu, “Wide field super-resolution surface imaging through plasmonic structured illumination microscopy,” Nano Lett. 14(8), 4634–4639 (2014).
[Crossref] [PubMed]

Wang, Q.

P. S. Tan, X. C. Yuan, G. H. Yuan, and Q. Wang, “High-resolution wide-field standing-wave surface plasmon resonance fluorescence microscopy with optical vortices,” Appl. Phys. Lett. 97(24), 241109 (2010).
[Crossref]

Wei, F.

J. L. Ponsetto, F. Wei, and Z. Liu, “Localized plasmon assisted structured illumination microscopy for wide-field high-speed dispersion-independent super resolution imaging,” Nanoscale 6(11), 5807–5812 (2014).
[Crossref] [PubMed]

F. Wei, D. Lu, H. Shen, W. Wan, J. L. Ponsetto, E. Huang, and Z. Liu, “Wide field super-resolution surface imaging through plasmonic structured illumination microscopy,” Nano Lett. 14(8), 4634–4639 (2014).
[Crossref] [PubMed]

Westphal, V.

V. Westphal, S. O. Rizzoli, M. A. Lauterbach, D. Kamin, R. Jahn, and S. W. Hell, “Video-rate far-field optical nanoscopy dissects synaptic vesicle movement,” Science 320(5873), 246–249 (2008).
[Crossref] [PubMed]

Wichmann, J.

Wilson, T.

C. J. R. Sheppard and T. Wilson, “The theory of the direct-view confocal microscope,” J. Microsc. 124(2), 107–117 (1981).
[Crossref] [PubMed]

Yuan, G. H.

P. S. Tan, X. C. Yuan, G. H. Yuan, and Q. Wang, “High-resolution wide-field standing-wave surface plasmon resonance fluorescence microscopy with optical vortices,” Appl. Phys. Lett. 97(24), 241109 (2010).
[Crossref]

Yuan, X. C.

P. S. Tan, X. C. Yuan, G. H. Yuan, and Q. Wang, “High-resolution wide-field standing-wave surface plasmon resonance fluorescence microscopy with optical vortices,” Appl. Phys. Lett. 97(24), 241109 (2010).
[Crossref]

Anal. Chem. (1)

N. Calander, “Theory and simulation of surface plasmon-coupled directional emission from fluorophores at planar structures,” Anal. Chem. 76(8), 2168–2173 (2004).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

P. S. Tan, X. C. Yuan, G. H. Yuan, and Q. Wang, “High-resolution wide-field standing-wave surface plasmon resonance fluorescence microscopy with optical vortices,” Appl. Phys. Lett. 97(24), 241109 (2010).
[Crossref]

Biophys. J. (1)

E. Chung, D. Kim, Y. Cui, Y. H. Kim, and P. T. So, “Two-dimensional standing wave total internal reflection fluorescence microscopy: superresolution imaging of single molecular and biological specimens,” Biophys. J. 93(5), 1747–1757 (2007).
[Crossref] [PubMed]

J. Microsc. (2)

C. J. R. Sheppard and T. Wilson, “The theory of the direct-view confocal microscope,” J. Microsc. 124(2), 107–117 (1981).
[Crossref] [PubMed]

M. G. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(2), 82–87 (2000).
[Crossref] [PubMed]

J. Opt. A (1)

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6(2), 259–268 (2004).
[Crossref]

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

Microsc. Acta (1)

C. Cremer and T. Cremer, “Considerations on a laser-scanning-microscope with high resolution and depth of field,” Microsc. Acta 81(1), 31–44 (1978).
[PubMed]

Nano Lett. (1)

F. Wei, D. Lu, H. Shen, W. Wan, J. L. Ponsetto, E. Huang, and Z. Liu, “Wide field super-resolution surface imaging through plasmonic structured illumination microscopy,” Nano Lett. 14(8), 4634–4639 (2014).
[Crossref] [PubMed]

Nanoscale (1)

J. L. Ponsetto, F. Wei, and Z. Liu, “Localized plasmon assisted structured illumination microscopy for wide-field high-speed dispersion-independent super resolution imaging,” Nanoscale 6(11), 5807–5812 (2014).
[Crossref] [PubMed]

Nature (1)

E. A. Ash and G. Nicholls, “Super-resolution aperture scanning microscope,” Nature 237(5357), 510–512 (1972).
[Crossref] [PubMed]

Opt. Lett. (2)

Proc. Natl. Acad. Sci. U.S.A. (1)

M. G. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. U.S.A. 102(37), 13081–13086 (2005).
[Crossref] [PubMed]

Proc. SPIE (1)

R. Heintzmann and C. G. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE 3568, 185–196 (1999).
[Crossref]

Science (1)

V. Westphal, S. O. Rizzoli, M. A. Lauterbach, D. Kamin, R. Jahn, and S. W. Hell, “Video-rate far-field optical nanoscopy dissects synaptic vesicle movement,” Science 320(5873), 246–249 (2008).
[Crossref] [PubMed]

Ultramicroscopy (1)

A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, “Development of a 500 Å spatial resolution light microscope,” Ultramicroscopy 13(3), 227–231 (1984).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 Tight-focusing configuration for the excitation of SP standing-waves with OVs. (a) The schematic diagram and (b)-(d) the calculated SP standing-waves excited by linearly-polarized OVs with topological charges of 1, 2 and 3.
Fig. 2
Fig. 2 SP-standing-wave patterns generated by the modified OV. A bow-tie shaped amplitude filter (a) was employed for shaping a full-intensity OV (b) to a bow-tie shaped intensity distribution (c). In the tightly focus configuration, the excitation position was cut to a smaller pair of arcs (green solid arcs in (d)). A standard SP-standing-wave pattern (e) with uniform periodicity could be generated by two counter-propagate SPs waves toward the silver film center. The successful excitation of SP waves could be confirmed by the dark lines (f) in the image obtained at the back focal plane.
Fig. 3
Fig. 3 Schematics of the PSIM system. A SLM, a half waveplate and an amplitude filter (θ = 20°) were used for dynamically controlling the phase (topological charges), polarization direction and amplitude of the incident OVs. The fluorescent beads were deposited onto the silver film. The emission light from the fluorescent beads was coupled back through the silver film and was collected via the same objective. Due to the SPCE phenomenon, the doughnut shaped PSFs were obtained. Insert: (a) computer generated hologram (CGH), (b) amplitude filter, (c) calculated intensity distribution of SP-Standing-wave pattern on silver film, (d) typical SPCE image of florescent beads.
Fig. 4
Fig. 4 Precise phase shifts of {-2π/3, 0, 2π/3} achieved by OVs with fractional topological charges. (a)-(c) Full-intensity distributions of OVs with topological charges {1, 1.66, and 2.34}. (a1)-(c1) Intensity distribution of modified OVs after amplitude filter. (a2)-(c2) Calculated intensity distribution of SP-standing-wave patterns excited by (a1)-(c1). (d) Intensity cross sections of (a2)-(c2).
Fig. 5
Fig. 5 Demonstration of PSIM system with sub-100nm resolution. (a) Original SPCE image with doughnut-shaped PSFs. (b) Image after R-L deconvolution algorithm processing. (c) Super-resolved image by applied reconstructed algorithm to three intermedia images. (d) The corresponding fluorescence intensity cross-section in (c).

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