Abstract

We present a series of experiments that demonstrate a super-sensitive chemical imaging technique based on multiphoton frequency-domain fluorescence lifetime imaging microscopy (MPM-FD-FLIM) that shows a 2× improvement in imaging speed compared to the theoretical limit of conventional MPM-FD-FLIM. Additionally, this technique produces unprecedented sensitivity over a large range of fluorescence lifetimes. These results are achieved through simple modifications to data analysis in a conventional MPM-FD-FLIM microscope and are based on an analytical model describing the signal-to-noise ratio (SNR) of a MPM-FD-FLIM system [J. Opt. Soc. Am. A 33, B1 (2016)]. Here we experimentally validate this model.

© 2016 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Investigation of signal-to-noise ratio in frequency-domain multiphoton fluorescence lifetime imaging microscopy

Yide Zhang, Aamir A. Khan, Genevieve D. Vigil, and Scott S. Howard
J. Opt. Soc. Am. A 33(7) B1-B11 (2016)

Generalized stepwise optical saturation enables super-resolution fluorescence lifetime imaging microscopy

Yide Zhang, David Benirschke, Ola Abdalsalam, and Scott S. Howard
Biomed. Opt. Express 9(9) 4077-4093 (2018)

Saturation-compensated measurements for fluorescence lifetime imaging microscopy

Yide Zhang, Genevieve D. Vigil, Lina Cao, Aamir A. Khan, David Benirschke, Tahsin Ahmed, Patrick Fay, and Scott S. Howard
Opt. Lett. 42(1) 155-158 (2017)

References

  • View by:
  • |
  • |
  • |

  1. J. Philip and K. Carlsson, “Theoretical investigation of the signal-to-noise ratio in fluorescence lifetime imaging,” J. Opt. Soc. Am. A 20, 368–379 (2003).
    [Crossref]
  2. D.-U. Li, J. Arlt, J. Richardson, R. Walker, A. Buts, D. Stoppa, E. Charbon, and R. Henderson, “Real-time fluorescence lifetime imaging system with a 32×32 0.13μm CMOS low dark-count single-photon avalanche diode array,” Opt. Express 18, 10257–10269 (2010).
    [Crossref] [PubMed]
  3. A. A. Khan, S. K. Fullerton-Shirey, and S. S. Howard, “Easily prepared ruthenium-complex nanomicelle probes for two-photon quantitative imaging of oxygen in aqueous media,” RSC Adv. 5, 291–300 (2015).
    [Crossref]
  4. S. Sakadzić, E. Roussakis, M. A. Yaseen, E. T. Mandeville, V. J. Srinivasan, K. Arai, S. Ruvinskaya, A. Devor, E. H. Lo, S. A. Vinogradov, and D. A. Boas, “Two-photon high-resolution measurement of partial pressure of oxygen in cerebral vasculature and tissue,” Nat. Methods 7, 755–759 (2010).
    [Crossref]
  5. J. Lecoq, A. Parpaleix, E. Roussakis, M. Ducros, Y. G. Houssen, S. A. Vinogradov, and S. Charpak, “Simultaneous two-photon imaging of oxygen and blood flow in deep cerebral vessels,” Nat. Med. 17, 893–898 (2011).
    [Crossref] [PubMed]
  6. M. A. Yaseen, S. Sakadžić, W. Wu, W. Becker, K. A. Kasischke, and D. A. Boas, “In vivo imaging of cerebral energy metabolism with two-photon fluorescence lifetime microscopy of NADH,” Biomed. Opt. Express 4, 307–321 (2013).
    [Crossref] [PubMed]
  7. S. S. Howard, A. Straub, N. G. Horton, D. Kobat, and C. Xu, “Frequency-multiplexed in vivo multiphoton phosphorescence lifetime microscopy,” Nat. Photonics 7, 33–37 (2012).
    [Crossref]
  8. W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
    [Crossref] [PubMed]
  9. W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21, 1369–1377 (2003).
    [Crossref] [PubMed]
  10. E. E. Hoover and J. A. Squier, “Advances in multiphoton microscopy technology,” Nat. Photonics 7, 93–101 (2013).
    [Crossref] [PubMed]
  11. S. Kumar, C. Dunsby, P. A. A. D. Beule, D. M. Owen, U. Anand, P. M. P. Lanigan, R. K. P. Benninger, D. M. Davis, M. A. A. Neil, P. Anand, C. Benham, A. Naylor, and P. M. W. French, “Multifocal multiphoton excitation and time correlated single photon counting detection for 3-D fluorescence lifetime imaging,” Opt. Express 15, 12548–12561 (2007).
    [Crossref] [PubMed]
  12. K. H. Kim, C. Buehler, K. Bahlmann, T. Ragan, W.-C. A. Lee, E. Nedivi, E. L. Heffer, S. Fantini, and P. T. C. So, “Multifocal multiphoton microscopy based on multianode photomultiplier tubes,” Opt. Express 15, 11658–11678 (2007).
    [Crossref] [PubMed]
  13. Y. Zhang, A. A. Khan, G. D. Vigil, and S. S. Howard, “Investigation of signal-to-noise ratio in frequency-domain multiphoton fluorescence lifetime imaging microscopy,” J. Opt. Soc. Am. A 33, B1–B11 (2016).
    [Crossref]
  14. A. D. Estrada and A. K. Dunn, “Improved sensitivity for two-photon frequency-domain lifetime measurement,” Opt. Express 18, 13631–13639 (2010).
    [Crossref] [PubMed]
  15. G. I. Redford and R. M. Clegg, “Polar plot representation for frequency-domain analysis of fluorescence lifetimes,” J. Fluoresc. 15, 805–815 (2005).
    [Crossref] [PubMed]

2016 (1)

2015 (1)

A. A. Khan, S. K. Fullerton-Shirey, and S. S. Howard, “Easily prepared ruthenium-complex nanomicelle probes for two-photon quantitative imaging of oxygen in aqueous media,” RSC Adv. 5, 291–300 (2015).
[Crossref]

2013 (2)

2012 (1)

S. S. Howard, A. Straub, N. G. Horton, D. Kobat, and C. Xu, “Frequency-multiplexed in vivo multiphoton phosphorescence lifetime microscopy,” Nat. Photonics 7, 33–37 (2012).
[Crossref]

2011 (1)

J. Lecoq, A. Parpaleix, E. Roussakis, M. Ducros, Y. G. Houssen, S. A. Vinogradov, and S. Charpak, “Simultaneous two-photon imaging of oxygen and blood flow in deep cerebral vessels,” Nat. Med. 17, 893–898 (2011).
[Crossref] [PubMed]

2010 (3)

2007 (2)

2005 (1)

G. I. Redford and R. M. Clegg, “Polar plot representation for frequency-domain analysis of fluorescence lifetimes,” J. Fluoresc. 15, 805–815 (2005).
[Crossref] [PubMed]

2003 (2)

J. Philip and K. Carlsson, “Theoretical investigation of the signal-to-noise ratio in fluorescence lifetime imaging,” J. Opt. Soc. Am. A 20, 368–379 (2003).
[Crossref]

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21, 1369–1377 (2003).
[Crossref] [PubMed]

1990 (1)

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[Crossref] [PubMed]

Anand, P.

Anand, U.

Arai, K.

S. Sakadzić, E. Roussakis, M. A. Yaseen, E. T. Mandeville, V. J. Srinivasan, K. Arai, S. Ruvinskaya, A. Devor, E. H. Lo, S. A. Vinogradov, and D. A. Boas, “Two-photon high-resolution measurement of partial pressure of oxygen in cerebral vasculature and tissue,” Nat. Methods 7, 755–759 (2010).
[Crossref]

Arlt, J.

Bahlmann, K.

Becker, W.

Benham, C.

Benninger, R. K. P.

Beule, P. A. A. D.

Boas, D. A.

M. A. Yaseen, S. Sakadžić, W. Wu, W. Becker, K. A. Kasischke, and D. A. Boas, “In vivo imaging of cerebral energy metabolism with two-photon fluorescence lifetime microscopy of NADH,” Biomed. Opt. Express 4, 307–321 (2013).
[Crossref] [PubMed]

S. Sakadzić, E. Roussakis, M. A. Yaseen, E. T. Mandeville, V. J. Srinivasan, K. Arai, S. Ruvinskaya, A. Devor, E. H. Lo, S. A. Vinogradov, and D. A. Boas, “Two-photon high-resolution measurement of partial pressure of oxygen in cerebral vasculature and tissue,” Nat. Methods 7, 755–759 (2010).
[Crossref]

Buehler, C.

Buts, A.

Carlsson, K.

Charbon, E.

Charpak, S.

J. Lecoq, A. Parpaleix, E. Roussakis, M. Ducros, Y. G. Houssen, S. A. Vinogradov, and S. Charpak, “Simultaneous two-photon imaging of oxygen and blood flow in deep cerebral vessels,” Nat. Med. 17, 893–898 (2011).
[Crossref] [PubMed]

Clegg, R. M.

G. I. Redford and R. M. Clegg, “Polar plot representation for frequency-domain analysis of fluorescence lifetimes,” J. Fluoresc. 15, 805–815 (2005).
[Crossref] [PubMed]

Davis, D. M.

Denk, W.

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[Crossref] [PubMed]

Devor, A.

S. Sakadzić, E. Roussakis, M. A. Yaseen, E. T. Mandeville, V. J. Srinivasan, K. Arai, S. Ruvinskaya, A. Devor, E. H. Lo, S. A. Vinogradov, and D. A. Boas, “Two-photon high-resolution measurement of partial pressure of oxygen in cerebral vasculature and tissue,” Nat. Methods 7, 755–759 (2010).
[Crossref]

Ducros, M.

J. Lecoq, A. Parpaleix, E. Roussakis, M. Ducros, Y. G. Houssen, S. A. Vinogradov, and S. Charpak, “Simultaneous two-photon imaging of oxygen and blood flow in deep cerebral vessels,” Nat. Med. 17, 893–898 (2011).
[Crossref] [PubMed]

Dunn, A. K.

Dunsby, C.

Estrada, A. D.

Fantini, S.

French, P. M. W.

Fullerton-Shirey, S. K.

A. A. Khan, S. K. Fullerton-Shirey, and S. S. Howard, “Easily prepared ruthenium-complex nanomicelle probes for two-photon quantitative imaging of oxygen in aqueous media,” RSC Adv. 5, 291–300 (2015).
[Crossref]

Heffer, E. L.

Henderson, R.

Hoover, E. E.

E. E. Hoover and J. A. Squier, “Advances in multiphoton microscopy technology,” Nat. Photonics 7, 93–101 (2013).
[Crossref] [PubMed]

Horton, N. G.

S. S. Howard, A. Straub, N. G. Horton, D. Kobat, and C. Xu, “Frequency-multiplexed in vivo multiphoton phosphorescence lifetime microscopy,” Nat. Photonics 7, 33–37 (2012).
[Crossref]

Houssen, Y. G.

J. Lecoq, A. Parpaleix, E. Roussakis, M. Ducros, Y. G. Houssen, S. A. Vinogradov, and S. Charpak, “Simultaneous two-photon imaging of oxygen and blood flow in deep cerebral vessels,” Nat. Med. 17, 893–898 (2011).
[Crossref] [PubMed]

Howard, S. S.

Y. Zhang, A. A. Khan, G. D. Vigil, and S. S. Howard, “Investigation of signal-to-noise ratio in frequency-domain multiphoton fluorescence lifetime imaging microscopy,” J. Opt. Soc. Am. A 33, B1–B11 (2016).
[Crossref]

A. A. Khan, S. K. Fullerton-Shirey, and S. S. Howard, “Easily prepared ruthenium-complex nanomicelle probes for two-photon quantitative imaging of oxygen in aqueous media,” RSC Adv. 5, 291–300 (2015).
[Crossref]

S. S. Howard, A. Straub, N. G. Horton, D. Kobat, and C. Xu, “Frequency-multiplexed in vivo multiphoton phosphorescence lifetime microscopy,” Nat. Photonics 7, 33–37 (2012).
[Crossref]

Kasischke, K. A.

Khan, A. A.

Y. Zhang, A. A. Khan, G. D. Vigil, and S. S. Howard, “Investigation of signal-to-noise ratio in frequency-domain multiphoton fluorescence lifetime imaging microscopy,” J. Opt. Soc. Am. A 33, B1–B11 (2016).
[Crossref]

A. A. Khan, S. K. Fullerton-Shirey, and S. S. Howard, “Easily prepared ruthenium-complex nanomicelle probes for two-photon quantitative imaging of oxygen in aqueous media,” RSC Adv. 5, 291–300 (2015).
[Crossref]

Kim, K. H.

Kobat, D.

S. S. Howard, A. Straub, N. G. Horton, D. Kobat, and C. Xu, “Frequency-multiplexed in vivo multiphoton phosphorescence lifetime microscopy,” Nat. Photonics 7, 33–37 (2012).
[Crossref]

Kumar, S.

Lanigan, P. M. P.

Lecoq, J.

J. Lecoq, A. Parpaleix, E. Roussakis, M. Ducros, Y. G. Houssen, S. A. Vinogradov, and S. Charpak, “Simultaneous two-photon imaging of oxygen and blood flow in deep cerebral vessels,” Nat. Med. 17, 893–898 (2011).
[Crossref] [PubMed]

Lee, W.-C. A.

Li, D.-U.

Lo, E. H.

S. Sakadzić, E. Roussakis, M. A. Yaseen, E. T. Mandeville, V. J. Srinivasan, K. Arai, S. Ruvinskaya, A. Devor, E. H. Lo, S. A. Vinogradov, and D. A. Boas, “Two-photon high-resolution measurement of partial pressure of oxygen in cerebral vasculature and tissue,” Nat. Methods 7, 755–759 (2010).
[Crossref]

Mandeville, E. T.

S. Sakadzić, E. Roussakis, M. A. Yaseen, E. T. Mandeville, V. J. Srinivasan, K. Arai, S. Ruvinskaya, A. Devor, E. H. Lo, S. A. Vinogradov, and D. A. Boas, “Two-photon high-resolution measurement of partial pressure of oxygen in cerebral vasculature and tissue,” Nat. Methods 7, 755–759 (2010).
[Crossref]

Naylor, A.

Nedivi, E.

Neil, M. A. A.

Owen, D. M.

Parpaleix, A.

J. Lecoq, A. Parpaleix, E. Roussakis, M. Ducros, Y. G. Houssen, S. A. Vinogradov, and S. Charpak, “Simultaneous two-photon imaging of oxygen and blood flow in deep cerebral vessels,” Nat. Med. 17, 893–898 (2011).
[Crossref] [PubMed]

Philip, J.

Ragan, T.

Redford, G. I.

G. I. Redford and R. M. Clegg, “Polar plot representation for frequency-domain analysis of fluorescence lifetimes,” J. Fluoresc. 15, 805–815 (2005).
[Crossref] [PubMed]

Richardson, J.

Roussakis, E.

J. Lecoq, A. Parpaleix, E. Roussakis, M. Ducros, Y. G. Houssen, S. A. Vinogradov, and S. Charpak, “Simultaneous two-photon imaging of oxygen and blood flow in deep cerebral vessels,” Nat. Med. 17, 893–898 (2011).
[Crossref] [PubMed]

S. Sakadzić, E. Roussakis, M. A. Yaseen, E. T. Mandeville, V. J. Srinivasan, K. Arai, S. Ruvinskaya, A. Devor, E. H. Lo, S. A. Vinogradov, and D. A. Boas, “Two-photon high-resolution measurement of partial pressure of oxygen in cerebral vasculature and tissue,” Nat. Methods 7, 755–759 (2010).
[Crossref]

Ruvinskaya, S.

S. Sakadzić, E. Roussakis, M. A. Yaseen, E. T. Mandeville, V. J. Srinivasan, K. Arai, S. Ruvinskaya, A. Devor, E. H. Lo, S. A. Vinogradov, and D. A. Boas, “Two-photon high-resolution measurement of partial pressure of oxygen in cerebral vasculature and tissue,” Nat. Methods 7, 755–759 (2010).
[Crossref]

Sakadzic, S.

S. Sakadzić, E. Roussakis, M. A. Yaseen, E. T. Mandeville, V. J. Srinivasan, K. Arai, S. Ruvinskaya, A. Devor, E. H. Lo, S. A. Vinogradov, and D. A. Boas, “Two-photon high-resolution measurement of partial pressure of oxygen in cerebral vasculature and tissue,” Nat. Methods 7, 755–759 (2010).
[Crossref]

Sakadžic, S.

So, P. T. C.

Squier, J. A.

E. E. Hoover and J. A. Squier, “Advances in multiphoton microscopy technology,” Nat. Photonics 7, 93–101 (2013).
[Crossref] [PubMed]

Srinivasan, V. J.

S. Sakadzić, E. Roussakis, M. A. Yaseen, E. T. Mandeville, V. J. Srinivasan, K. Arai, S. Ruvinskaya, A. Devor, E. H. Lo, S. A. Vinogradov, and D. A. Boas, “Two-photon high-resolution measurement of partial pressure of oxygen in cerebral vasculature and tissue,” Nat. Methods 7, 755–759 (2010).
[Crossref]

Stoppa, D.

Straub, A.

S. S. Howard, A. Straub, N. G. Horton, D. Kobat, and C. Xu, “Frequency-multiplexed in vivo multiphoton phosphorescence lifetime microscopy,” Nat. Photonics 7, 33–37 (2012).
[Crossref]

Strickler, J. H.

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[Crossref] [PubMed]

Vigil, G. D.

Vinogradov, S. A.

J. Lecoq, A. Parpaleix, E. Roussakis, M. Ducros, Y. G. Houssen, S. A. Vinogradov, and S. Charpak, “Simultaneous two-photon imaging of oxygen and blood flow in deep cerebral vessels,” Nat. Med. 17, 893–898 (2011).
[Crossref] [PubMed]

S. Sakadzić, E. Roussakis, M. A. Yaseen, E. T. Mandeville, V. J. Srinivasan, K. Arai, S. Ruvinskaya, A. Devor, E. H. Lo, S. A. Vinogradov, and D. A. Boas, “Two-photon high-resolution measurement of partial pressure of oxygen in cerebral vasculature and tissue,” Nat. Methods 7, 755–759 (2010).
[Crossref]

Walker, R.

Webb, W. W.

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21, 1369–1377 (2003).
[Crossref] [PubMed]

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[Crossref] [PubMed]

Williams, R. M.

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21, 1369–1377 (2003).
[Crossref] [PubMed]

Wu, W.

Xu, C.

S. S. Howard, A. Straub, N. G. Horton, D. Kobat, and C. Xu, “Frequency-multiplexed in vivo multiphoton phosphorescence lifetime microscopy,” Nat. Photonics 7, 33–37 (2012).
[Crossref]

Yaseen, M. A.

M. A. Yaseen, S. Sakadžić, W. Wu, W. Becker, K. A. Kasischke, and D. A. Boas, “In vivo imaging of cerebral energy metabolism with two-photon fluorescence lifetime microscopy of NADH,” Biomed. Opt. Express 4, 307–321 (2013).
[Crossref] [PubMed]

S. Sakadzić, E. Roussakis, M. A. Yaseen, E. T. Mandeville, V. J. Srinivasan, K. Arai, S. Ruvinskaya, A. Devor, E. H. Lo, S. A. Vinogradov, and D. A. Boas, “Two-photon high-resolution measurement of partial pressure of oxygen in cerebral vasculature and tissue,” Nat. Methods 7, 755–759 (2010).
[Crossref]

Zhang, Y.

Zipfel, W. R.

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21, 1369–1377 (2003).
[Crossref] [PubMed]

Biomed. Opt. Express (1)

J. Fluoresc. (1)

G. I. Redford and R. M. Clegg, “Polar plot representation for frequency-domain analysis of fluorescence lifetimes,” J. Fluoresc. 15, 805–815 (2005).
[Crossref] [PubMed]

J. Opt. Soc. Am. A (2)

Nat. Biotechnol. (1)

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21, 1369–1377 (2003).
[Crossref] [PubMed]

Nat. Med. (1)

J. Lecoq, A. Parpaleix, E. Roussakis, M. Ducros, Y. G. Houssen, S. A. Vinogradov, and S. Charpak, “Simultaneous two-photon imaging of oxygen and blood flow in deep cerebral vessels,” Nat. Med. 17, 893–898 (2011).
[Crossref] [PubMed]

Nat. Methods (1)

S. Sakadzić, E. Roussakis, M. A. Yaseen, E. T. Mandeville, V. J. Srinivasan, K. Arai, S. Ruvinskaya, A. Devor, E. H. Lo, S. A. Vinogradov, and D. A. Boas, “Two-photon high-resolution measurement of partial pressure of oxygen in cerebral vasculature and tissue,” Nat. Methods 7, 755–759 (2010).
[Crossref]

Nat. Photonics (2)

E. E. Hoover and J. A. Squier, “Advances in multiphoton microscopy technology,” Nat. Photonics 7, 93–101 (2013).
[Crossref] [PubMed]

S. S. Howard, A. Straub, N. G. Horton, D. Kobat, and C. Xu, “Frequency-multiplexed in vivo multiphoton phosphorescence lifetime microscopy,” Nat. Photonics 7, 33–37 (2012).
[Crossref]

Opt. Express (4)

RSC Adv. (1)

A. A. Khan, S. K. Fullerton-Shirey, and S. S. Howard, “Easily prepared ruthenium-complex nanomicelle probes for two-photon quantitative imaging of oxygen in aqueous media,” RSC Adv. 5, 291–300 (2015).
[Crossref]

Science (1)

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[Crossref] [PubMed]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1 MPM-FD-FLIM experimental setup.
Fig. 2
Fig. 2 (a) Representative experimental lifetime temporal plots (left) and histograms (right) for the 1ω, 2ω, and DC&1ω methods. (b) Experimental (symbols) and analytical (curves, from [13]) values of F versus modulation frequency for the 1.6 μs [Ru(dpp)3]2+ nanomi-celle sample. Inset: zoomed scale.
Fig. 3
Fig. 3 (a) Measured (symbols) and linear fit (lines) lifetime relative error versus integration time. Dashed green line: theoretical limit for one-photon sinusoidal FD-FLIM. (b) Lifetime standard deviation versus fluorescence lifetime for a constant modulation frequency of 62.5 kHz. Inset: zoomed view, clearly showing the expanded DC&1ω frequency range.

Equations (3)

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

a k = 1 T 0 T e 2 ( t ) exp ( i k ω t ) d t , d k = 1 T 0 T p ( t ) exp ( i k ω t ) d t , k = 0 , ± 1 , ± 2 , .
G 0 = m 2 + 2 2 , G 1 = m ω τ 1 + ω 2 τ 2 , G 2 = m 2 4 1 1 + 4 ω 2 τ 2 , S 0 = 0 , S 1 = m 1 1 + ω 2 τ 2 , S 2 = m 2 4 2 ω τ 1 + 4 ω 2 τ 2 .
τ = 1 ω 2 m m 2 + 2 G 0 S 1 1 .

Metrics