Abstract

Stimulated-emission depletion (STED) microscopy improves image resolution by encoding additional spatial information in a second stimulated-decay channel with a spatially-varying strength. Here we demonstrate that spatial information is also encoded in the fluorophore lifetime and that this information can be used to improve the spatial resolution of STED microscopy. By solving a kinetic model for emission in the presence of a time-varying STED pulse, we derive the effective resolution as a function of fluorophore lifetime and pulse duration. We find that the best resolution for a given pulse power is achieved with a pulse of infinitesimally short duration; however, the maximum resolution can be restored for pulses of finite duration by time-gating the fluorescence signal. In parallel, we consider time-gating in the presence of a continuous-wave (CW) STED beam and find that time-gating produces theoretically unbounded resolution with finite laser power. In both cases, the cost of this improved resolution is a reduction in the brightness of the final image. We conclude by discussing situations in which time-gated STED microscopy (T-STED) may provide improved microscope performance beyond an increase in resolution.

© 2011 OSA

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2010 (2)

G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, “Superresolution imaging using single-molecule localization,” Annu. Rev. Phys. Chem. 61(1), 345–367 (2010).
[CrossRef] [PubMed]

M. Leutenegger, C. Eggeling, and S. W. Hell, “Analytical description of STED microscopy performance,” Opt. Express 18(25), 26417–26429 (2010).
[CrossRef] [PubMed]

2009 (5)

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(3), 144–147 (2009).
[CrossRef]

B. Huang, M. Bates, and X. W. Zhuang, “Super-resolution fluorescence microscopy,” Annu. Rev. Biochem. 78(1), 993–1016 (2009).
[CrossRef] [PubMed]

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

S. R. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Natl. Acad. Sci. U.S.A. 106(9), 2995–2999 (2009).
[CrossRef] [PubMed]

K. Y. Han, K. I. Willig, E. Rittweger, F. Jelezko, C. Eggeling, and S. W. Hell, “Three-dimensional stimulated emission depletion microscopy of nitrogen-vacancy centers in diamond using continuous-wave light,” Nano Lett. 9(9), 3323–3329 (2009).
[CrossRef] [PubMed]

2008 (4)

2007 (1)

K. I. Willig, B. Harke, R. Medda, and S. W. Hell, “STED microscopy with continuous wave beams,” Nat. Methods 4(11), 915–918 (2007).
[CrossRef] [PubMed]

2006 (3)

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(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

S. T. Hess, T. P. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91(11), 4258–4272 (2006).
[CrossRef] [PubMed]

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–796 (2006).
[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]

2003 (2)

2002 (1)

M. Dyba and S. W. Hell, “Focal spots of size lambda/23 open up far-field fluorescence microscopy at 33 nm axial resolution,” Phys. Rev. Lett. 88(16), 163901 (2002).
[CrossRef] [PubMed]

1999 (1)

M. G. Gustafsson, D. A. Agard, and J. W. Sedat, “I5M: 3D widefield light microscopy with better than 100 nm axial resolution,” J. Microsc. 195(1), 10–16 (1999).
[CrossRef] [PubMed]

1994 (1)

Agard, D. A.

M. G. Gustafsson, D. A. Agard, and J. W. Sedat, “I5M: 3D widefield light microscopy with better than 100 nm axial resolution,” J. Microsc. 195(1), 10–16 (1999).
[CrossRef] [PubMed]

Auksorius, E.

Bates, M.

B. Huang, M. Bates, and X. W. Zhuang, “Super-resolution fluorescence microscopy,” Annu. Rev. Biochem. 78(1), 993–1016 (2009).
[CrossRef] [PubMed]

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–796 (2006).
[CrossRef] [PubMed]

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(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Biteen, J. S.

S. R. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Natl. Acad. Sci. U.S.A. 106(9), 2995–2999 (2009).
[CrossRef] [PubMed]

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(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Boruah, B. R.

Davidson, M.

G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, “Superresolution imaging using single-molecule localization,” Annu. Rev. Phys. Chem. 61(1), 345–367 (2010).
[CrossRef] [PubMed]

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(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Dunsby, C.

Dyba, M.

M. Dyba and S. W. Hell, “Photostability of a fluorescent marker under pulsed excited-state depletion through stimulated emission,” Appl. Opt. 42(25), 5123–5129 (2003).
[CrossRef] [PubMed]

M. Dyba and S. W. Hell, “Focal spots of size lambda/23 open up far-field fluorescence microscopy at 33 nm axial resolution,” Phys. Rev. Lett. 88(16), 163901 (2002).
[CrossRef] [PubMed]

Eggeling, C.

M. Leutenegger, C. Eggeling, and S. W. Hell, “Analytical description of STED microscopy performance,” Opt. Express 18(25), 26417–26429 (2010).
[CrossRef] [PubMed]

K. Y. Han, K. I. Willig, E. Rittweger, F. Jelezko, C. Eggeling, and S. W. Hell, “Three-dimensional stimulated emission depletion microscopy of nitrogen-vacancy centers in diamond using continuous-wave light,” Nano Lett. 9(9), 3323–3329 (2009).
[CrossRef] [PubMed]

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(3), 144–147 (2009).
[CrossRef]

French, P. M. W.

Girirajan, T. P.

S. T. Hess, T. P. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91(11), 4258–4272 (2006).
[CrossRef] [PubMed]

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, D. A. Agard, and J. W. Sedat, “I5M: 3D widefield light microscopy with better than 100 nm axial resolution,” J. Microsc. 195(1), 10–16 (1999).
[CrossRef] [PubMed]

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(3), 144–147 (2009).
[CrossRef]

K. Y. Han, K. I. Willig, E. Rittweger, F. Jelezko, C. Eggeling, and S. W. Hell, “Three-dimensional stimulated emission depletion microscopy of nitrogen-vacancy centers in diamond using continuous-wave light,” Nano Lett. 9(9), 3323–3329 (2009).
[CrossRef] [PubMed]

Harke, B.

B. Harke, J. Keller, C. K. Ullal, V. Westphal, A. Schönle, and S. W. Hell, “Resolution scaling in STED microscopy,” Opt. Express 16(6), 4154–4162 (2008).
[CrossRef] [PubMed]

K. I. Willig, B. Harke, R. Medda, and S. W. Hell, “STED microscopy with continuous wave beams,” Nat. Methods 4(11), 915–918 (2007).
[CrossRef] [PubMed]

Hell, S. W.

M. Leutenegger, C. Eggeling, and S. W. Hell, “Analytical description of STED microscopy performance,” Opt. Express 18(25), 26417–26429 (2010).
[CrossRef] [PubMed]

K. Y. Han, K. I. Willig, E. Rittweger, F. Jelezko, C. Eggeling, and S. W. Hell, “Three-dimensional stimulated emission depletion microscopy of nitrogen-vacancy centers in diamond using continuous-wave light,” Nano Lett. 9(9), 3323–3329 (2009).
[CrossRef] [PubMed]

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(3), 144–147 (2009).
[CrossRef]

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

B. Harke, J. Keller, C. K. Ullal, V. Westphal, A. Schönle, and S. W. Hell, “Resolution scaling in STED microscopy,” Opt. Express 16(6), 4154–4162 (2008).
[CrossRef] [PubMed]

B. R. Rankin, R. R. Kellner, and S. W. Hell, “Stimulated-emission-depletion microscopy with a multicolor stimulated-Raman-scattering light source,” Opt. Lett. 33(21), 2491–2493 (2008).
[PubMed]

K. I. Willig, B. Harke, R. Medda, and S. W. Hell, “STED microscopy with continuous wave beams,” Nat. Methods 4(11), 915–918 (2007).
[CrossRef] [PubMed]

M. Dyba and S. W. Hell, “Photostability of a fluorescent marker under pulsed excited-state depletion through stimulated emission,” Appl. Opt. 42(25), 5123–5129 (2003).
[CrossRef] [PubMed]

S. W. Hell, “Toward fluorescence nanoscopy,” Nat. Biotechnol. 21(11), 1347–1355 (2003).
[CrossRef] [PubMed]

M. Dyba and S. W. Hell, “Focal spots of size lambda/23 open up far-field fluorescence microscopy at 33 nm axial resolution,” Phys. Rev. Lett. 88(16), 163901 (2002).
[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]

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(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Hess, S. T.

S. T. Hess, T. P. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91(11), 4258–4272 (2006).
[CrossRef] [PubMed]

Huang, B.

B. Huang, M. Bates, and X. W. Zhuang, “Super-resolution fluorescence microscopy,” Annu. Rev. Biochem. 78(1), 993–1016 (2009).
[CrossRef] [PubMed]

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(3), 144–147 (2009).
[CrossRef]

Jelezko, F.

K. Y. Han, K. I. Willig, E. Rittweger, F. Jelezko, C. Eggeling, and S. W. Hell, “Three-dimensional stimulated emission depletion microscopy of nitrogen-vacancy centers in diamond using continuous-wave light,” Nano Lett. 9(9), 3323–3329 (2009).
[CrossRef] [PubMed]

Keller, J.

Kellner, R. R.

Kennedy, G.

Lakowicz, J. R.

J. R. Lakowicz and B. R. Masters, “Principles of fluorescence spectroscopy,” J. Biomed. Opt. 13(2), 029901 (2008).
[CrossRef]

Lanigan, P. M. P.

Leutenegger, M.

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(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Lippincott-Schwartz, J.

G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, “Superresolution imaging using single-molecule localization,” Annu. Rev. Phys. Chem. 61(1), 345–367 (2010).
[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(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Liu, N.

S. R. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Natl. Acad. Sci. U.S.A. 106(9), 2995–2999 (2009).
[CrossRef] [PubMed]

Lord, S. J.

S. R. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Natl. Acad. Sci. U.S.A. 106(9), 2995–2999 (2009).
[CrossRef] [PubMed]

Manley, S.

G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, “Superresolution imaging using single-molecule localization,” Annu. Rev. Phys. Chem. 61(1), 345–367 (2010).
[CrossRef] [PubMed]

Mason, M. D.

S. T. Hess, T. P. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91(11), 4258–4272 (2006).
[CrossRef] [PubMed]

Masters, B. R.

J. R. Lakowicz and B. R. Masters, “Principles of fluorescence spectroscopy,” J. Biomed. Opt. 13(2), 029901 (2008).
[CrossRef]

Medda, R.

K. I. Willig, B. Harke, R. Medda, and S. W. Hell, “STED microscopy with continuous wave beams,” Nat. Methods 4(11), 915–918 (2007).
[CrossRef] [PubMed]

Moerner, W. E.

S. R. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Natl. Acad. Sci. U.S.A. 106(9), 2995–2999 (2009).
[CrossRef] [PubMed]

Neil, M. A. A.

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(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Patterson, G.

G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, “Superresolution imaging using single-molecule localization,” Annu. Rev. Phys. Chem. 61(1), 345–367 (2010).
[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(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Pavani, S. R.

S. R. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Natl. Acad. Sci. U.S.A. 106(9), 2995–2999 (2009).
[CrossRef] [PubMed]

Piestun, R.

S. R. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Natl. Acad. Sci. U.S.A. 106(9), 2995–2999 (2009).
[CrossRef] [PubMed]

Rankin, B. R.

Rittweger, E.

K. Y. Han, K. I. Willig, E. Rittweger, F. Jelezko, C. Eggeling, and S. W. Hell, “Three-dimensional stimulated emission depletion microscopy of nitrogen-vacancy centers in diamond using continuous-wave light,” Nano Lett. 9(9), 3323–3329 (2009).
[CrossRef] [PubMed]

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(3), 144–147 (2009).
[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(10), 793–796 (2006).
[CrossRef] [PubMed]

Schönle, A.

Sedat, J. W.

M. G. Gustafsson, D. A. Agard, and J. W. Sedat, “I5M: 3D widefield light microscopy with better than 100 nm axial resolution,” J. Microsc. 195(1), 10–16 (1999).
[CrossRef] [PubMed]

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(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Thompson, M. A.

S. R. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Natl. Acad. Sci. U.S.A. 106(9), 2995–2999 (2009).
[CrossRef] [PubMed]

Twieg, R. J.

S. R. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Natl. Acad. Sci. U.S.A. 106(9), 2995–2999 (2009).
[CrossRef] [PubMed]

Ullal, C. K.

Westphal, V.

Wichmann, J.

Willig, K. I.

K. Y. Han, K. I. Willig, E. Rittweger, F. Jelezko, C. Eggeling, and S. W. Hell, “Three-dimensional stimulated emission depletion microscopy of nitrogen-vacancy centers in diamond using continuous-wave light,” Nano Lett. 9(9), 3323–3329 (2009).
[CrossRef] [PubMed]

K. I. Willig, B. Harke, R. Medda, and S. W. Hell, “STED microscopy with continuous wave beams,” Nat. Methods 4(11), 915–918 (2007).
[CrossRef] [PubMed]

Zhuang, X.

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Figures (4)

Fig. 1
Fig. 1

Fluorophore Lifetimes in the Presence of a Stimulating Beam. (a) The intensity distribution for the stimulating beam. (b) The probability of emitting in the fluorescent channel (solid) when a uniform field of excited fluorophores (dashed) are exposed to a STED beam with tS = τ and S = 23. (c) The number of fluorophores that have emitted via the spontaneous emission channel as a function of time, normalized by the total number of fluorophores that will emit in this channel (N = NF (t = ∞)).) The different colors represent different locations along the STED beam corresponding to different probabilities of emitting via spontaneous emission (1, blue; 1/2, green; 1/3, red; 1/5, orange). These positions are circled in panel (b). (d) The fraction of fluorophores that will emit by spontaneous emission but have not yet done so at a given time. All distances are measured in units of spatial parameter of the STED beam, ω, and all times are measured in units of the unstimulated fluorophore lifetime, τ.

Fig. 2
Fig. 2

Improving Spatial Resolution with Time-Gating. (a) Timing diagram for pulsed STED with time-gated detection. Immediately following an excitation pulse of negligible duration (blue), a STED pulse of duration tS begins (red). The final image is created from the spontaneous emission that arrives during a time window starting tG after the completion of one excitation pulse and lasting until the beginning of the next (green). (b) The probability of emitting via normal fluorescence as a function of position (black) when a uniform field of excited fluorophores (dashed black) are exposed to the pulsed STED beam described in Fig. 1. The probability of emitting via spontaneous emission after the STED pulse has completed (red, dashed), and this probability corrected for the decrease in image brightness (red, solid). (c) The PSFs for confocal detection (black), normal STED microscopy (red), or time-gated STED microscopy (blue). Probabilities and PSF were calculated with a STED strength sufficient to produce a 2-fold decrease in the FWHM of the effective PSF compared to the confocal PSF and a STED pulse duration equal to the unstimulated fluorophore lifetime, i.e. S = 23 and tS = τ. All distances are measured in units of the 1/e waist of the electric field of the excitation beam. The T-STED PSF in panel (c) has been corrected for the decrease in image brightness.

Fig. 3
Fig. 3

T-STED Resolution for Pulsed STED. (a) The N-fold increase in the resolving power of a STED microscope over that of a confocal microscope as a function of the strength of the STED beam, S. Different pulse durations are plotted in different colors (instant; black; 0.5τ, red; τ, blue.) Increase in resolving power is judged by the relative decrease of the FWHM of the effective PSF from that of the confocal PSF. (b) T-STED resolution improvement over regular STED microscopy judged by the decrease in either the FWHM (black) or the FWQM (blue) for different STED beam strengths. Different pulsed durations are plotted (α = 0.1, large dash; α = 0.5; small dash; and α = 1.0, solid.) (c) Resolution improvement of T-STED over pulsed STED for different α determined with the FWHM (black) or the FWQM (blue). Different STED beam strengths are plotted corresponding to the STED strength required to produce a 2-fold (large dash,) 5-fold (small dash,) or 20-fold (solid) improvement in the FWHM with an instantaneous pulse (black trace in panel a). FWHM and FWQM values were calculated numerically from Eqs. (11) and (18).

Fig. 4
Fig. 4

CW T-STED. (a) Timing diagram for time-gating with a CW STED beam. (b) The effective PSF for different CW T-STED parameters. Plotted are the confocal PSF (black); the effective PSF with a CW STED beam with STED power sufficient to reduce the FWHM to 1/2 that of the confocal PSF (red; S = σγω2τ = 16.5); the same STED beam with tG = 0.5τ (blue); tG = τ (green); tG = 2τ (purple); and tG = 4τ (brown). (c) The effective PSF corrected for the decrease in image brightness. The color scheme is the same as in panel (b). (d) The N-fold improvement in the FWHM of the effective PSF with respect to the confocal PSF versus strength of the STED beam. tG = 0,1,2,4τ are plotted as solid black, dashed red, dashed blue, and dashed green respectively. (e) The N-fold improvement in the FWHM due to CW T-STED as compared to the confocal PSF versus time-gate duration for different STED beam strengths. Red, blue, and green correspond to STED strengths that decrease the FWHM by 2, 3, and 4-fold in the absence of a time-gate. All distances are measured in units of the 1/e waist of the electric field of the excitation beam. FWHM values were calculated numerically from Eqs. (11) and (23).

Equations (25)

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d n ( r , t ) d t = k n ( r , t ) σ I ( r ) ( 1 θ ( t t S ) ) n ( r , t )
d N S ( r , t ) d t = σ I ( r ) ( 1 θ ( t t S ) ) n ( r , t ) ,
d N F ( r , t ) d t = k n ( r , t ) .
n ( r , t ) = n 0 { e ( k + σ I ( r ) ) t e σ I ( r ) t S e k t : : t t S t > t S
N F ( r , t ) = n 0 { k k + σ I ( r ) ( 1 e ( k + σ I ( r ) ) t ) t t S k k + σ I ( r ) + σ I ( r ) k + σ I ( r ) e ( k + σ I ( r ) ) t S e σ I ( r ) t S e k t t > t S
N S ( r , t ) = n 0 { σ I ( r ) k + σ I ( r ) ( 1 e ( k + σ I ( r ) ) t ) t t S σ I ( r ) k + σ I ( r ) ( 1 e ( k + σ I ( r ) ) t S ) t > t S .
lim t N F ( r , t ) / n 0 η S T E D ( r ) = k k + σ I ( r ) + σ I ( r ) k + σ I ( r ) e ( k + σ I ( r ) ) t S
lim t N S ( r , t ) / n 0 = σ I ( r ) k + σ I ( r ) ( 1 e ( k + σ I ( r ) ) t S ) .
I ( r ) = γ r 2 e 2 r 2 / ω 2
S = σ γ ω 2 t S = σ ε .
P S F C ( r ) ~ e 4 r 2 / ω 2
P S F S T E D ( r ) = P S F C ( r ) η S T E D ( r ) = e 4 r 2 / ω 2 ( k k + σ I ( r ) + σ I ( r ) k + σ I ( r ) e ( k + σ I ( r ) ) t S ) .
α = t S / τ
E ( r ) = σ I ( r ) t S
η S T E D ( r ) = ( α α + E ( r ) + E ( r ) α + E ( r ) e ( α + E ( r ) ) ) .
lim α 0 η S T E D ( r ) = e σ I ( r ) t S = e E ( r ) .
η T S T E D ( r ; t G ) = lim t N F ( t ) N F ( t G ) n 0 = ( k k + σ I ( r ) e ( k + σ I ( r ) ) t G + σ I ( r ) k + σ I ( r ) e ( k + σ I ( r ) ) t S ) .
η T S T E D ( r ; t G = t S ) = e k t S e σ I ( r ) t S .
ω e f f / ω = ( 4 + S ( t G t S + τ t S ( 1 e t S τ ( 1 t G t S ) ) ) ) 1 / 2 .
ω e f f / ω = ( 4 + S τ t S ( 1 e t S / τ ) ) 1 / 2 .
ω e f f / ω = ( 4 + S ) 1 / 2 .
N F ( r , t ) = n 0 k k + σ I ( r ) ( 1 e ( k + σ I ( r ) ) t )
lim t N F ( r , t ) = n 0 k k + σ I ( r ) .
η C W T S T E D ( r ; t G ) = N F ( r , ) N F ( r , t G ) n 0 = k k + σ I ( r ) e k t G e σ I ( r ) t G .
ω e f f / ω = ( 4 + σ γ ω 2 τ ( 1 + t G / τ ) ) 1 / 2 = ( 4 + S ( 1 + t G / τ ) ) 1 / 2 .

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