Abstract

We present a novel super-resolution fluorescence lifetime microscopy technique called generalized stepwise optical saturation (GSOS) that generalizes and extends the concept of the recently demonstrated stepwise optical saturation (SOS) super-resolution microscopy [Biomed. Opt. Express 9, 1613 (2018)]. The theoretical basis of GSOS is developed based on exploring the dynamics of a two-level fluorophore model and using perturbation theory. We show that although both SOS and GSOS utilize the linear combination of M raw images to increase the imaging resolution by a factor of M, SOS is a special and the simplest case of GSOS. The super-resolution capability is demonstrated with theoretical analysis and numerical simulations for GSOS with sinusoidal and pulse-train modulations. Using GSOS with pulse-train modulation, super-resolution and fluorescence lifetime imaging microscopy (FLIM) images can be obtained simultaneously. The super-resolution FLIM capability is experimentally demonstrated with a cell sample on a custom-built two-photon frequency-domain (FD) FLIM system based on radio frequency analog signal processing. To our knowledge, this is the first implementation of super-resolution imaging in FD-FLIM.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (1)

2017 (3)

T. Hato, S. Winfree, R. Day, R. M. Sandoval, B. A. Molitoris, M. C. Yoder, R. C. Wiggins, Y. Zheng, K. W. Dunn, and P. C. Dagher, “Two-photon intravital fluorescence lifetime imaging of the kidney reveals cell-type specific metabolic signatures,” J. Am. Soc. Nephrol. 28, 2420–2430 (2017).
[Crossref] [PubMed]

Y. Zhang, G. D. Vigil, L. Cao, A. A. Khan, D. Benirschke, T. Ahmed, P. Fay, and S. S. Howard, “Saturation-compensated measurements for fluorescence lifetime imaging microscopy,” Opt. Lett. 42, 155–158 (2017).
[Crossref] [PubMed]

F. Görlitz, D. Corcoran, E. Garcia Castano, B. Leitinger, M. Neil, C. Dunsby, and P. French, “Mapping molecular function to biological nanostructure: combining structured illumination microscopy with fluorescence lifetime imaging (SIM + FLIM),” Photonics 4, 40 (2017).
[Crossref]

2016 (1)

2014 (1)

A. S. Kristoffersen, S. R. Erga, B. Hamre, and Ø. Frette, “Testing fluorescence lifetime standards using two-photon excitation and time-domain instrumentation: rhodamine B, coumarin 6 and lucifer yellow,” J. Fluoresc. 24, 1015–1024 (2014).
[Crossref] [PubMed]

2012 (1)

M. D. Lesoine, S. Bose, J. W. Petrich, and E. A. Smith, “Supercontinuum stimulated emission depletion fluorescence lifetime imaging,” J. Phys. Chem. B 116, 7821–7826 (2012).
[Crossref] [PubMed]

2011 (2)

Y. Sun, R. N. Day, and A. Periasamy, “Investigating protein-protein interactions in living cells using fluorescence lifetime imaging microscopy,” Nat. Protoc. 6, 1324–1340 (2011).
[Crossref] [PubMed]

J. Bückers, D. Wildanger, G. Vicidomini, L. Kastrup, and S. W. Hell, “Simultaneous multi-lifetime multi-color STED imaging for colocalization analyses,” Opt. Express 19, 3130–3143 (2011).
[Crossref] [PubMed]

2009 (1)

J. Humpolíčková, A. Benda, and J. Enderlein, “Optical saturation as a versatile tool to enhance resolution in confocal microscopy,” Biophys. J. 97, 2623–2629 (2009).
[Crossref]

2008 (1)

2007 (4)

K. Fujita, M. Kobayashi, S. Kawano, M. Yamanaka, and S. Kawata, “High-resolution confocal microscopy by saturated excitation of fluorescence,” Phys. Rev. Lett. 99, 228105 (2007).
[Crossref]

C.-W. Chang, D. Sud, and M.-A. Mycek, “Fluorescence lifetime imaging microscopy,” Method. Cell Biol. 81, 495–524 (2007).
[Crossref]

S. W. Hell, “Far-field optical nanoscopy,” Science 316, 1153–1158 (2007).
[Crossref] [PubMed]

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. VandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem. 79, 2137–2149 (2007).
[Crossref] [PubMed]

2006 (3)

R. M. Zucker, “Quality assessment of confocal microscopy slide based systems: Performance,” Cytometry Part A 69A, 659–676 (2006).
[Crossref]

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, 1642–1645 (2006).
[Crossref] [PubMed]

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

2005 (2)

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

C. Eggeling, A. Volkmer, and C. A. M. Seidel, “Molecular photobleaching kinetics of rhodamine 6G by one- and two-photon induced confocal fluorescence microscopy,” Chem Phys Chem 6, 791–804 (2005).
[Crossref] [PubMed]

2003 (1)

2001 (1)

1996 (1)

1994 (1)

Agronskaia, A. V.

H. C. Gerritsen, A. Draaijer, D. J. van den Heuvel, and A. V. Agronskaia, “Fluorescence lifetime imaging in scanning microscopy,” in “Handbook Of Biological Confocal Microscopy,” J. Pawley, ed. (Springer US, Boston, MA, 2006), pp. 516–534, 3rd ed.
[Crossref]

Ahmed, T.

Ameloot, M.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. VandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem. 79, 2137–2149 (2007).
[Crossref] [PubMed]

Auksorius, E.

Basaric, N.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. VandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem. 79, 2137–2149 (2007).
[Crossref] [PubMed]

Bates, M.

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

Benda, A.

J. Humpolíčková, A. Benda, and J. Enderlein, “Optical saturation as a versatile tool to enhance resolution in confocal microscopy,” Biophys. J. 97, 2623–2629 (2009).
[Crossref]

Benirschke, D.

Y. Zhang, G. D. Vigil, L. Cao, A. A. Khan, D. Benirschke, T. Ahmed, P. Fay, and S. S. Howard, “Saturation-compensated measurements for fluorescence lifetime imaging microscopy,” Opt. Lett. 42, 155–158 (2017).
[Crossref] [PubMed]

Y. Zhang, D. Benirschke, and S. S. Howard, “Stepwise optical saturation microscopy: obtaining super-resolution images with conventional fluorescence microscopes,” in “Biophotonics Congress: Biomedical Optics Congress 2018,” (OSA, 2018), p. JTh3A.27.

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, 1642–1645 (2006).
[Crossref] [PubMed]

Boens, N.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. VandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem. 79, 2137–2149 (2007).
[Crossref] [PubMed]

Boerckel, J. D.

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, 1642–1645 (2006).
[Crossref] [PubMed]

Boruah, B. R.

Bose, S.

M. D. Lesoine, S. Bose, J. W. Petrich, and E. A. Smith, “Supercontinuum stimulated emission depletion fluorescence lifetime imaging,” J. Phys. Chem. B 116, 7821–7826 (2012).
[Crossref] [PubMed]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic Press, 2003).

Bückers, J.

Cao, L.

Carlsson, K.

Chang, C.-W.

C.-W. Chang, D. Sud, and M.-A. Mycek, “Fluorescence lifetime imaging microscopy,” Method. Cell Biol. 81, 495–524 (2007).
[Crossref]

Corcoran, D.

F. Görlitz, D. Corcoran, E. Garcia Castano, B. Leitinger, M. Neil, C. Dunsby, and P. French, “Mapping molecular function to biological nanostructure: combining structured illumination microscopy with fluorescence lifetime imaging (SIM + FLIM),” Photonics 4, 40 (2017).
[Crossref]

Dagher, P. C.

T. Hato, S. Winfree, R. Day, R. M. Sandoval, B. A. Molitoris, M. C. Yoder, R. C. Wiggins, Y. Zheng, K. W. Dunn, and P. C. Dagher, “Two-photon intravital fluorescence lifetime imaging of the kidney reveals cell-type specific metabolic signatures,” J. Am. Soc. Nephrol. 28, 2420–2430 (2017).
[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, 1642–1645 (2006).
[Crossref] [PubMed]

Day, R.

T. Hato, S. Winfree, R. Day, R. M. Sandoval, B. A. Molitoris, M. C. Yoder, R. C. Wiggins, Y. Zheng, K. W. Dunn, and P. C. Dagher, “Two-photon intravital fluorescence lifetime imaging of the kidney reveals cell-type specific metabolic signatures,” J. Am. Soc. Nephrol. 28, 2420–2430 (2017).
[Crossref] [PubMed]

Day, R. N.

Y. Sun, R. N. Day, and A. Periasamy, “Investigating protein-protein interactions in living cells using fluorescence lifetime imaging microscopy,” Nat. Protoc. 6, 1324–1340 (2011).
[Crossref] [PubMed]

De Grauw, C. J.

Draaijer, A.

H. C. Gerritsen, A. Draaijer, D. J. van den Heuvel, and A. V. Agronskaia, “Fluorescence lifetime imaging in scanning microscopy,” in “Handbook Of Biological Confocal Microscopy,” J. Pawley, ed. (Springer US, Boston, MA, 2006), pp. 516–534, 3rd ed.
[Crossref]

Dunn, K. W.

T. Hato, S. Winfree, R. Day, R. M. Sandoval, B. A. Molitoris, M. C. Yoder, R. C. Wiggins, Y. Zheng, K. W. Dunn, and P. C. Dagher, “Two-photon intravital fluorescence lifetime imaging of the kidney reveals cell-type specific metabolic signatures,” J. Am. Soc. Nephrol. 28, 2420–2430 (2017).
[Crossref] [PubMed]

Dunsby, C.

F. Görlitz, D. Corcoran, E. Garcia Castano, B. Leitinger, M. Neil, C. Dunsby, and P. French, “Mapping molecular function to biological nanostructure: combining structured illumination microscopy with fluorescence lifetime imaging (SIM + FLIM),” Photonics 4, 40 (2017).
[Crossref]

E. Auksorius, B. R. Boruah, C. Dunsby, P. M. P. Lanigan, G. Kennedy, M. A. A. Neil, and P. M. W. French, “Stimulated emission depletion microscopy with a supercontinuum source and fluorescence lifetime imaging,” Opt. Lett. 33, 113–115 (2008).
[Crossref] [PubMed]

Eggeling, C.

C. Eggeling, A. Volkmer, and C. A. M. Seidel, “Molecular photobleaching kinetics of rhodamine 6G by one- and two-photon induced confocal fluorescence microscopy,” Chem Phys Chem 6, 791–804 (2005).
[Crossref] [PubMed]

Enderlein, J.

J. Humpolíčková, A. Benda, and J. Enderlein, “Optical saturation as a versatile tool to enhance resolution in confocal microscopy,” Biophys. J. 97, 2623–2629 (2009).
[Crossref]

Engelborghs, Y.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. VandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem. 79, 2137–2149 (2007).
[Crossref] [PubMed]

Erga, S. R.

A. S. Kristoffersen, S. R. Erga, B. Hamre, and Ø. Frette, “Testing fluorescence lifetime standards using two-photon excitation and time-domain instrumentation: rhodamine B, coumarin 6 and lucifer yellow,” J. Fluoresc. 24, 1015–1024 (2014).
[Crossref] [PubMed]

Fay, P.

French, P.

F. Görlitz, D. Corcoran, E. Garcia Castano, B. Leitinger, M. Neil, C. Dunsby, and P. French, “Mapping molecular function to biological nanostructure: combining structured illumination microscopy with fluorescence lifetime imaging (SIM + FLIM),” Photonics 4, 40 (2017).
[Crossref]

French, P. M. W.

Frette, Ø.

A. S. Kristoffersen, S. R. Erga, B. Hamre, and Ø. Frette, “Testing fluorescence lifetime standards using two-photon excitation and time-domain instrumentation: rhodamine B, coumarin 6 and lucifer yellow,” J. Fluoresc. 24, 1015–1024 (2014).
[Crossref] [PubMed]

Fujita, K.

K. Fujita, M. Kobayashi, S. Kawano, M. Yamanaka, and S. Kawata, “High-resolution confocal microscopy by saturated excitation of fluorescence,” Phys. Rev. Lett. 99, 228105 (2007).
[Crossref]

Garcia Castano, E.

F. Görlitz, D. Corcoran, E. Garcia Castano, B. Leitinger, M. Neil, C. Dunsby, and P. French, “Mapping molecular function to biological nanostructure: combining structured illumination microscopy with fluorescence lifetime imaging (SIM + FLIM),” Photonics 4, 40 (2017).
[Crossref]

Gerritsen, H. C.

C. J. De Grauw and H. C. Gerritsen, “Multiple time-gate module for fluorescence lifetime imaging,” Appl. Spectrosc. 55, 670–678 (2001).
[Crossref]

H. C. Gerritsen, A. Draaijer, D. J. van den Heuvel, and A. V. Agronskaia, “Fluorescence lifetime imaging in scanning microscopy,” in “Handbook Of Biological Confocal Microscopy,” J. Pawley, ed. (Springer US, Boston, MA, 2006), pp. 516–534, 3rd ed.
[Crossref]

Görlitz, F.

F. Görlitz, D. Corcoran, E. Garcia Castano, B. Leitinger, M. Neil, C. Dunsby, and P. French, “Mapping molecular function to biological nanostructure: combining structured illumination microscopy with fluorescence lifetime imaging (SIM + FLIM),” Photonics 4, 40 (2017).
[Crossref]

Gratton, E.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. VandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem. 79, 2137–2149 (2007).
[Crossref] [PubMed]

Gryczynski, I.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. VandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem. 79, 2137–2149 (2007).
[Crossref] [PubMed]

Gustafsson, M. G. L.

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

Hamre, B.

A. S. Kristoffersen, S. R. Erga, B. Hamre, and Ø. Frette, “Testing fluorescence lifetime standards using two-photon excitation and time-domain instrumentation: rhodamine B, coumarin 6 and lucifer yellow,” J. Fluoresc. 24, 1015–1024 (2014).
[Crossref] [PubMed]

Hato, T.

T. Hato, S. Winfree, R. Day, R. M. Sandoval, B. A. Molitoris, M. C. Yoder, R. C. Wiggins, Y. Zheng, K. W. Dunn, and P. C. Dagher, “Two-photon intravital fluorescence lifetime imaging of the kidney reveals cell-type specific metabolic signatures,” J. Am. Soc. Nephrol. 28, 2420–2430 (2017).
[Crossref] [PubMed]

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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, 1642–1645 (2006).
[Crossref] [PubMed]

Hofkens, J.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. VandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem. 79, 2137–2149 (2007).
[Crossref] [PubMed]

Howard, S. S.

Humpolícková, J.

J. Humpolíčková, A. Benda, and J. Enderlein, “Optical saturation as a versatile tool to enhance resolution in confocal microscopy,” Biophys. J. 97, 2623–2629 (2009).
[Crossref]

Kastrup, L.

Kawano, S.

K. Fujita, M. Kobayashi, S. Kawano, M. Yamanaka, and S. Kawata, “High-resolution confocal microscopy by saturated excitation of fluorescence,” Phys. Rev. Lett. 99, 228105 (2007).
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Kawata, S.

K. Fujita, M. Kobayashi, S. Kawano, M. Yamanaka, and S. Kawata, “High-resolution confocal microscopy by saturated excitation of fluorescence,” Phys. Rev. Lett. 99, 228105 (2007).
[Crossref]

Kennedy, G.

Khan, A. A.

Kobayashi, M.

K. Fujita, M. Kobayashi, S. Kawano, M. Yamanaka, and S. Kawata, “High-resolution confocal microscopy by saturated excitation of fluorescence,” Phys. Rev. Lett. 99, 228105 (2007).
[Crossref]

Krajcarski, D. T.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. VandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem. 79, 2137–2149 (2007).
[Crossref] [PubMed]

Kristoffersen, A. S.

A. S. Kristoffersen, S. R. Erga, B. Hamre, and Ø. Frette, “Testing fluorescence lifetime standards using two-photon excitation and time-domain instrumentation: rhodamine B, coumarin 6 and lucifer yellow,” J. Fluoresc. 24, 1015–1024 (2014).
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S. R. Kurtz, “Mixers as phase detectors,” Watkins-Johnson Tech-Notes5 (1978).

Lakowicz, J. R.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. VandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem. 79, 2137–2149 (2007).
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Lanigan, P. M. P.

Lefèvre, J.-P.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. VandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem. 79, 2137–2149 (2007).
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Leitinger, B.

F. Görlitz, D. Corcoran, E. Garcia Castano, B. Leitinger, M. Neil, C. Dunsby, and P. French, “Mapping molecular function to biological nanostructure: combining structured illumination microscopy with fluorescence lifetime imaging (SIM + FLIM),” Photonics 4, 40 (2017).
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M. D. Lesoine, S. Bose, J. W. Petrich, and E. A. Smith, “Supercontinuum stimulated emission depletion fluorescence lifetime imaging,” J. Phys. Chem. B 116, 7821–7826 (2012).
[Crossref] [PubMed]

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, 1642–1645 (2006).
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Lippincott-Schwartz, J.

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, 1642–1645 (2006).
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Malak, H.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. VandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem. 79, 2137–2149 (2007).
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Mason, D. E.

Miura, A.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. VandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem. 79, 2137–2149 (2007).
[Crossref] [PubMed]

Molitoris, B. A.

T. Hato, S. Winfree, R. Day, R. M. Sandoval, B. A. Molitoris, M. C. Yoder, R. C. Wiggins, Y. Zheng, K. W. Dunn, and P. C. Dagher, “Two-photon intravital fluorescence lifetime imaging of the kidney reveals cell-type specific metabolic signatures,” J. Am. Soc. Nephrol. 28, 2420–2430 (2017).
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C.-W. Chang, D. Sud, and M.-A. Mycek, “Fluorescence lifetime imaging microscopy,” Method. Cell Biol. 81, 495–524 (2007).
[Crossref]

Nallathamby, P. D.

Neil, M.

F. Görlitz, D. Corcoran, E. Garcia Castano, B. Leitinger, M. Neil, C. Dunsby, and P. French, “Mapping molecular function to biological nanostructure: combining structured illumination microscopy with fluorescence lifetime imaging (SIM + FLIM),” Photonics 4, 40 (2017).
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O’Connor, D.

D. O’Connor, Time-Correlated Single Photon Counting (Academic Press, 2012).

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, 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. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref] [PubMed]

Periasamy, A.

Y. Sun, R. N. Day, and A. Periasamy, “Investigating protein-protein interactions in living cells using fluorescence lifetime imaging microscopy,” Nat. Protoc. 6, 1324–1340 (2011).
[Crossref] [PubMed]

Petrich, J. W.

M. D. Lesoine, S. Bose, J. W. Petrich, and E. A. Smith, “Supercontinuum stimulated emission depletion fluorescence lifetime imaging,” J. Phys. Chem. B 116, 7821–7826 (2012).
[Crossref] [PubMed]

Philip, J.

Phillips, D.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. VandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem. 79, 2137–2149 (2007).
[Crossref] [PubMed]

Pouget, J.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. VandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem. 79, 2137–2149 (2007).
[Crossref] [PubMed]

Qin, W.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. VandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem. 79, 2137–2149 (2007).
[Crossref] [PubMed]

Roeder, R. K.

Rumbles, G.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. VandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem. 79, 2137–2149 (2007).
[Crossref] [PubMed]

Rust, M. J.

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

Sandoval, R. M.

T. Hato, S. Winfree, R. Day, R. M. Sandoval, B. A. Molitoris, M. C. Yoder, R. C. Wiggins, Y. Zheng, K. W. Dunn, and P. C. Dagher, “Two-photon intravital fluorescence lifetime imaging of the kidney reveals cell-type specific metabolic signatures,” J. Am. Soc. Nephrol. 28, 2420–2430 (2017).
[Crossref] [PubMed]

Seidel, C. A. M.

C. Eggeling, A. Volkmer, and C. A. M. Seidel, “Molecular photobleaching kinetics of rhodamine 6G by one- and two-photon induced confocal fluorescence microscopy,” Chem Phys Chem 6, 791–804 (2005).
[Crossref] [PubMed]

Sillen, A.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. VandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem. 79, 2137–2149 (2007).
[Crossref] [PubMed]

Silva, N. D.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. VandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem. 79, 2137–2149 (2007).
[Crossref] [PubMed]

Smith, E. A.

M. D. Lesoine, S. Bose, J. W. Petrich, and E. A. Smith, “Supercontinuum stimulated emission depletion fluorescence lifetime imaging,” J. Phys. Chem. B 116, 7821–7826 (2012).
[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, 1642–1645 (2006).
[Crossref] [PubMed]

Sud, D.

C.-W. Chang, D. Sud, and M.-A. Mycek, “Fluorescence lifetime imaging microscopy,” Method. Cell Biol. 81, 495–524 (2007).
[Crossref]

Sun, Y.

Y. Sun, R. N. Day, and A. Periasamy, “Investigating protein-protein interactions in living cells using fluorescence lifetime imaging microscopy,” Nat. Protoc. 6, 1324–1340 (2011).
[Crossref] [PubMed]

Szabo, A. G.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. VandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem. 79, 2137–2149 (2007).
[Crossref] [PubMed]

Tamai, N.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. VandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem. 79, 2137–2149 (2007).
[Crossref] [PubMed]

Valeur, B.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. VandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem. 79, 2137–2149 (2007).
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van den Heuvel, D. J.

H. C. Gerritsen, A. Draaijer, D. J. van den Heuvel, and A. V. Agronskaia, “Fluorescence lifetime imaging in scanning microscopy,” in “Handbook Of Biological Confocal Microscopy,” J. Pawley, ed. (Springer US, Boston, MA, 2006), pp. 516–534, 3rd ed.
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van Hoek, A.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. VandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem. 79, 2137–2149 (2007).
[Crossref] [PubMed]

VandeVen, M.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. VandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem. 79, 2137–2149 (2007).
[Crossref] [PubMed]

Vicidomini, G.

Vigil, G. D.

Visser, A. J. W. G.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. VandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem. 79, 2137–2149 (2007).
[Crossref] [PubMed]

Volkmer, A.

C. Eggeling, A. Volkmer, and C. A. M. Seidel, “Molecular photobleaching kinetics of rhodamine 6G by one- and two-photon induced confocal fluorescence microscopy,” Chem Phys Chem 6, 791–804 (2005).
[Crossref] [PubMed]

Webb, W. W.

Wichmann, J.

Wiggins, R. C.

T. Hato, S. Winfree, R. Day, R. M. Sandoval, B. A. Molitoris, M. C. Yoder, R. C. Wiggins, Y. Zheng, K. W. Dunn, and P. C. Dagher, “Two-photon intravital fluorescence lifetime imaging of the kidney reveals cell-type specific metabolic signatures,” J. Am. Soc. Nephrol. 28, 2420–2430 (2017).
[Crossref] [PubMed]

Wildanger, D.

Willaert, K.

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. VandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem. 79, 2137–2149 (2007).
[Crossref] [PubMed]

Winfree, S.

T. Hato, S. Winfree, R. Day, R. M. Sandoval, B. A. Molitoris, M. C. Yoder, R. C. Wiggins, Y. Zheng, K. W. Dunn, and P. C. Dagher, “Two-photon intravital fluorescence lifetime imaging of the kidney reveals cell-type specific metabolic signatures,” J. Am. Soc. Nephrol. 28, 2420–2430 (2017).
[Crossref] [PubMed]

Xu, C.

Yamanaka, M.

K. Fujita, M. Kobayashi, S. Kawano, M. Yamanaka, and S. Kawata, “High-resolution confocal microscopy by saturated excitation of fluorescence,” Phys. Rev. Lett. 99, 228105 (2007).
[Crossref]

Yoder, M. C.

T. Hato, S. Winfree, R. Day, R. M. Sandoval, B. A. Molitoris, M. C. Yoder, R. C. Wiggins, Y. Zheng, K. W. Dunn, and P. C. Dagher, “Two-photon intravital fluorescence lifetime imaging of the kidney reveals cell-type specific metabolic signatures,” J. Am. Soc. Nephrol. 28, 2420–2430 (2017).
[Crossref] [PubMed]

Zhang, Y.

Zheng, Y.

T. Hato, S. Winfree, R. Day, R. M. Sandoval, B. A. Molitoris, M. C. Yoder, R. C. Wiggins, Y. Zheng, K. W. Dunn, and P. C. Dagher, “Two-photon intravital fluorescence lifetime imaging of the kidney reveals cell-type specific metabolic signatures,” J. Am. Soc. Nephrol. 28, 2420–2430 (2017).
[Crossref] [PubMed]

Zhuang, X.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3, 793–796 (2006).
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R. M. Zucker, “Quality assessment of confocal microscopy slide based systems: Performance,” Cytometry Part A 69A, 659–676 (2006).
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Anal. Chem. (1)

N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. VandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem. 79, 2137–2149 (2007).
[Crossref] [PubMed]

Appl. Spectrosc. (1)

Biomed. Opt. Express (1)

Biophys. J. (1)

J. Humpolíčková, A. Benda, and J. Enderlein, “Optical saturation as a versatile tool to enhance resolution in confocal microscopy,” Biophys. J. 97, 2623–2629 (2009).
[Crossref]

Chem Phys Chem (1)

C. Eggeling, A. Volkmer, and C. A. M. Seidel, “Molecular photobleaching kinetics of rhodamine 6G by one- and two-photon induced confocal fluorescence microscopy,” Chem Phys Chem 6, 791–804 (2005).
[Crossref] [PubMed]

Cytometry Part A (1)

R. M. Zucker, “Quality assessment of confocal microscopy slide based systems: Performance,” Cytometry Part A 69A, 659–676 (2006).
[Crossref]

J. Am. Soc. Nephrol. (1)

T. Hato, S. Winfree, R. Day, R. M. Sandoval, B. A. Molitoris, M. C. Yoder, R. C. Wiggins, Y. Zheng, K. W. Dunn, and P. C. Dagher, “Two-photon intravital fluorescence lifetime imaging of the kidney reveals cell-type specific metabolic signatures,” J. Am. Soc. Nephrol. 28, 2420–2430 (2017).
[Crossref] [PubMed]

J. Fluoresc. (1)

A. S. Kristoffersen, S. R. Erga, B. Hamre, and Ø. Frette, “Testing fluorescence lifetime standards using two-photon excitation and time-domain instrumentation: rhodamine B, coumarin 6 and lucifer yellow,” J. Fluoresc. 24, 1015–1024 (2014).
[Crossref] [PubMed]

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

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

J. Phys. Chem. B (1)

M. D. Lesoine, S. Bose, J. W. Petrich, and E. A. Smith, “Supercontinuum stimulated emission depletion fluorescence lifetime imaging,” J. Phys. Chem. B 116, 7821–7826 (2012).
[Crossref] [PubMed]

Method. Cell Biol. (1)

C.-W. Chang, D. Sud, and M.-A. Mycek, “Fluorescence lifetime imaging microscopy,” Method. Cell Biol. 81, 495–524 (2007).
[Crossref]

Nat. Methods (1)

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3, 793–796 (2006).
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Figures (5)

Fig. 1
Fig. 1 Simulated results for GSOS microscopy with sinusoidal modulation. (a) Complex trajectories and (b) magnitude and phase of fluorescence harmonics qk with different excitation irradiances. (c) Magnitude and (d) phase of complex PSFs of diffraction-limited (DL), two-step, and three-step GSOS microscopy.
Fig. 2
Fig. 2 Simulated results for GSOS microscopy with pulse-train modulation. (a) Complex trajectories and (b) magnitude and phase of fluorescence harmonics qk with different excitation irradiances. (c) Magnitude and (d) phase of complex PSFs of diffraction-limited, two-step, and three-step GSOS microscopy.
Fig. 3
Fig. 3 Block diagram of the lifetime measurement method.
Fig. 4
Fig. 4 Data acquisition and processing steps in two-step GSOS.
Fig. 5
Fig. 5 Two-step GSOS microscopy generates super-resolution fluorescence lifetime images of fixed BPAE cells labeled with MitoTracker Red CMXRos (mitochondria), Alexa Fluor 488 phalloidin (F-actin), and DAPI (nuclei). (a) Magnitude and phase of the fundamental harmonic images of the first step (q1,1), the second step (q1,1), and their linear combination result (q1,2−GSOS) in a two-step GSOS microscope. (b) Magnitude and phase profiles along the yellow lines in (a) showing that two-step GSOS is able to provide super-resolved magnitude images as well as preserve their lifetime (phase) information. (c) Composite images of the diffraction-limited (q1,1) and two-step GSOS (q1,2−GSOS) harmonics, where the harmonics’ magnitudes and the fluorescence lifetimes are mapped to the pixels’ brightness and hue, respectively.

Tables (1)

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Table 1 Coefficients for the linear combination in SOS and GSOS microscopy.

Equations (44)

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d F ( t ) d t = K N 0 g p σ N γ N I N ( t ) ( g p σ N γ N I N ( t ) + 1 τ ) F ( t ) ,
F ( t ) = F ( 0 ) + λ F ( 1 ) + λ 2 F ( 2 ) + λ 3 F ( 3 ) + .
λ 0 : d F ( 0 ) ( t ) d t = 1 τ F ( 0 ) ( t ) ,
λ 1 : d F ( 1 ) ( t ) d t = K N 0 g p σ N γ N I N ( t ) g p σ N γ N I N ( t ) F ( 0 ) ( t ) 1 τ F ( 1 ) ( t ) ,
λ 2 : d F ( 2 ) ( t ) d t = g p σ N γ N I N ( t ) F ( 1 ) ( t ) 1 τ F ( 2 ) ( t ) ,
λ 3 : d F ( 3 ) ( t ) d t = g p σ N γ N I N ( t ) F ( 2 ) ( t ) 1 τ F ( 3 ) ( t ) ,
λ n : d F ( n ) ( t ) d t = g p σ N γ N I N ( t ) F ( n 1 ) ( t ) 1 τ F ( n ) ( t ) , ( n 2 ) .
I N ( t ) = I ¯ N k = p k exp ( i k ω t ) , p k = 1 T 0 T I N ( t ) I ¯ N exp ( i k ω t ) d t ,
F ( t ) = k = q k exp ( i k ω t ) , q k = 1 T 0 T F ( t ) exp ( i k ω t ) d t ,
q k = q k ( 0 ) + λ q k ( 1 ) + λ 2 q k ( 2 ) + λ 3 q k ( 3 ) + ,
F ( n ) ( t ) = k = q k ( n ) exp ( i k ω t ) , q k ( n ) = 1 T 0 T F ( n ) ( t ) exp ( i k ω t ) d t .
d F ( 1 ) ( t ) d t = K N 0 g p σ N γ N I N ( t ) 1 τ F ( 1 ) ( t ) .
l = ( q l ( 1 ) 1 + i l ω τ τ ) exp ( i l ω t ) = K N 0 g p σ N γ N I ¯ N m = p m exp ( i m ω t ) .
q k ( 1 ) = K N 0 a I ¯ N ( 1 1 + i k ω τ p k ) .
l = ( q l ( 2 ) 1 + i l ω τ τ ) exp ( i l ω t ) = g p σ N γ N I ¯ N m = l = p m q l ( 1 ) exp ( i ( l + m ) ω t ) .
q k ( 2 ) = τ 1 + i k ω τ g p σ N γ N I ¯ N [ p k * q k ( 1 ) ] .
q k ( 2 ) = K N 0 a 2 I ¯ 2 N ( 1 1 + i k ω τ [ p k * ( 1 1 + i k ω τ p k ) ] ) .
l = ( q l ( 3 ) 1 + i l ω τ τ ) exp ( i l ω t ) = g p σ N γ N I ¯ N m = l = p m q l ( 2 ) exp ( i ( l + m ) ω t ) .
q k ( 3 ) = τ 1 + i k ω τ g p σ N γ N I ¯ N [ p k * q k ( 2 ) ] .
q k ( 3 ) = K N 0 a 3 I ¯ 3 N ( 1 1 + i k ω τ [ p k * ( 1 1 + i k ω τ [ p k * ( 1 1 + i k ω τ p k ) ] ) ] ) ,
l = ( q l ( n ) 1 + i l ω τ τ ) exp ( i l ω t ) = g p σ N γ N I ¯ N m = l = p m q l ( n 1 ) exp ( i ( l + m ) ω t ) .
q k ( n ) = τ 1 + i k ω τ g p σ N γ N I ¯ N [ p k * q k ( n 1 ) ] .
q k ( n ) = K N 0 ( 1 ) n + 1 a n I ¯ n N ( 1 1 + i k ω τ [ p k * ( 1 1 + i k ω τ [ p k * ( ) ] ) ] ) .
q k = q k ( 1 ) + q k ( 2 ) + q k ( 3 ) + ,
q k ( x ) = K N 0 { a I 0 N g N ( x ) ( 1 1 + i k ω τ p k ) a 2 I 0 2 N g 2 N ( x ) ( 1 1 + i k ω τ [ p k * ( 1 1 + i k ω τ p k ) ] ) + a 3 I 0 3 N g 3 N ( x ) ( 1 1 + i k ω τ [ p k * ( 1 1 + i k ω τ [ p k * ( 1 1 + i k ω τ p k ) ] ) ] ) } .
q k , m ( x ) = K N 0 { a I 0 m N g N ( x ) ( 1 1 + i k ω τ p k ) a 2 I 0 m 2 N g 2 N ( x ) ( 1 1 + i k ω τ [ p k * ( 1 1 + i k ω τ p k ) ] ) + a 3 I 0 m 3 N g 3 N ( x ) ( 1 1 + i k ω τ [ p k * ( 1 1 + i k ω τ [ p k * ( 1 1 + i k ω τ p k ) ] ) ] ) } .
q k , M GSOS ( x ) = m = 1 M c m q k , m ( x ) ,
q k , 2 GSOS ( x ) = c 1 q k , 1 ( x ) + c 2 q k , 2 ( x ) = K N 0 { a 2 I 01 N ( I 01 N I 02 N ) g 2 g ( x ) ( 1 1 + i k ω τ [ p k * ( 1 1 + i k ω τ p k ) ] ) + a 3 I 01 N ( I 01 2 N I 02 2 N ) g 3 N ( x ) × ( 1 1 + i k ω τ [ p k * ( 1 1 + i k ω τ [ p k * ( 1 1 + i k ω τ ) ] ) ] ) } .
F ( x ) = q 0 ( x ) = K N 0 { a I 0 N g N ( x ) a 2 I 0 N g 2 N ( x ) + a 3 I 0 3 N g 3 N ( x ) } ,
q k ( 1 ) = K N 0 a I ¯ N 1 1 + i k ω τ .
q k ( 2 ) = a I ¯ N 1 1 + i k ω τ [ p k * q k ( 1 ) ] ,
[ p k * q k ( 1 ) ] = j = p k j q j ( 1 ) = j = q j ( 1 ) = K N 0 a I ¯ N j = 1 1 + i j ω τ .
j = 1 1 + i j ω τ = 1 + m = 1 2 1 + ( m ω τ ) 2 ,
q k ( 2 ) = K N 0 a 2 I ¯ 2 N R 1 1 + i k ω τ .
q k ( 3 ) = a I ¯ N 1 1 + i k ω τ [ p k * q k ( 2 ) ] = a I ¯ N 1 1 + i k ω τ j = q j ( 2 ) = K N 0 a 3 I ¯ 3 N R 1 1 + i k ω τ j = 1 1 + i k ω τ = K N 0 a 3 I ¯ 3 N R 2 1 1 + i k ω τ .
q k ( n ) = K N 0 ( 1 ) n + 1 a n I ¯ n N R n 1 1 1 + i k ω τ .
q k = K N 0 1 1 + i k ω τ n = 1 ( 1 ) n + 1 a n I n N R n 1 = K N 0 1 1 + i k ω τ ( a I ¯ N a 2 I ¯ 2 N R + a 3 I ¯ 3 N R 2 ) .
| p k | = K N 0 1 1 + ( k ω τ ) 2 ( a I ¯ N a 2 I ¯ 2 N R + a 3 I ¯ 3 N R 2 ) .
p k = p k , M GSOS = arctan ( k ω τ ) ,
τ = 1 k ω tan ( p k ) = 1 k ω tan ( p k , M GSOS ) .
F = q 0 = K N 0 ( a I ¯ N a 2 I ¯ 2 N + a 3 I ¯ 3 N ) ,
V IF ( φ ) = 1 T 0 T v IF ( t , φ ) d t = G [ Re { q 1 } sin φ Im { q 1 } cos ( φ ) ] + D .
q 1 = arctan [ V IF ( π ) V IF ( 0 ) V IF ( 0.5 π ) V IF ( 1.5 π ) ] ,
| q 1 | [ V IF ( π ) V IF ( 0 ) ] 2 + [ V IF ( 0.5 π ) V IF ( 1.5 π ) ] 2 ,