Abstract

In this Letter, we theoretically describe photoacoustic signal generation of molecules, for which triplet relaxation can be neglected, by considering the excited state lifetime, the fluorescence quantum yield, and the fast vibrational relaxation. We show that the phase response of the photoacoustic signal can be exploited to determine the excited state lifetime of dark molecules. For fluorescent molecules, the phase response can be used to determine the fluorescence quantum yield directly without the need of reference samples.

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References

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  1. A. G. Bell, Am. J. Sci. s3-20, 305 (1880).
    [Crossref]
  2. C. G. A. Hoelen and F. F. M. de Mul, J. Acoust. Soc. Am. 106, 695 (1999).
    [Crossref]
  3. I. G. Calasso, W. Craig, and G. J. Diebold, Phys. Rev. Lett. 86, 3550 (2001).
    [Crossref]
  4. L. V. Wang, IEEE J. Sel. Top. Quantum Electron. 14, 171 (2008).
    [Crossref]
  5. J. Märk, F.-J. Schmitt, and J. Laufer, J. Opt. 18, 054009 (2016).
    [Crossref]
  6. A. Forbrich, P. Shao, W. Shi, and R. J. Zemp, J. Opt. 18, 124001 (2016).
    [Crossref]
  7. L. J. Rothberg, M. Bernstein, and K. S. Peters, J. Chem. Phys. 79, 2569 (1983).
    [Crossref]
  8. W. Keller, W. Schubert, R. Germer, and E. Strauss, J. Phys. Colloq. 44, C6-397 (1983).
    [Crossref]
  9. J. R. Lakowicz, H. Szmacinski, K. Nowaczyk, K. W. Berndt, and M. Johnson, Anal. Biochem. 202, 316 (1992).
    [Crossref]
  10. C. Würth, M. G. González, R. Niessner, U. Panne, C. Haisch, and U. R. Genger, Talanta 90, 30 (2012).
    [Crossref]
  11. M. Kasha, Discuss. Faraday Soc. 9, 14 (1950).
    [Crossref]
  12. D. Reiser and A. Laubereau, Appl. Phys. B 27, 115 (1982).
    [Crossref]
  13. P. Troeger, C.-H. Liu, and A. Laubereau, Time Resolved Vibrational Spectroscopy, Springer Proceedings in Physics (Springer, 1985), pp. 62–66.
  14. A. M. Winkler, K. Maslov, and L. V. Wang, J. Biomed. Opt. 18, 097003 (2013).
    [Crossref]
  15. J. Yao and L. V. Wang, Photoacoustics 2, 87 (2014).
    [Crossref]
  16. J. Märk, F.-J. Schmitt, C. Theiss, H. Dortay, T. Friedrich, and J. Laufer, Biomed. Opt. Express 6, 2522 (2015).
    [Crossref]
  17. G. Langer, B. Buchegger, J. Jacak, T. A. Klar, and T. Berer, Biomed. Opt. Express 7, 2692 (2016).
    [Crossref]
  18. ATTO-TEC GmbH, “ATTO-TEC fluorescent labels and dyes product catalogue 2016/2018” (2016).
  19. P. Harms, J. Sipior, N. Ram, G. M. Carter, and G. Rao, Rev. Sci. Instrum. 70, 1535 (1999).
    [Crossref]
  20. J. Olmsted, J. Phys. Chem. 83, 2581 (1979).
    [Crossref]
  21. K. Suzuki, A. Kobayashi, S. Kaneko, K. Takehira, T. Yoshihara, H. Ishida, Y. Shiina, S. Oishi, and S. Tobita, Phys. Chem. Chem. Phys. 11, 9850 (2009).
    [Crossref]
  22. A. M. Brouwer, Pure Appl. Chem. 83, 2213 (2011).
    [Crossref]
  23. S. Fery-Forgues and D. Lavabre, J. Chem. Educ. 76, 1260 (1999).
    [Crossref]
  24. G. Langer and T. Berer, Proc. SPIE 10494, 1049465 (2018).
    [Crossref]
  25. C. Zhang, K. Maslov, and L. V. Wang, Opt. Lett. 35, 3195 (2010).
    [Crossref]
  26. E. M. Strohm, M. J. Moore, and M. C. Kolios, Photoacoustics 4, 36 (2016).
    [Crossref]

2018 (1)

G. Langer and T. Berer, Proc. SPIE 10494, 1049465 (2018).
[Crossref]

2016 (4)

E. M. Strohm, M. J. Moore, and M. C. Kolios, Photoacoustics 4, 36 (2016).
[Crossref]

G. Langer, B. Buchegger, J. Jacak, T. A. Klar, and T. Berer, Biomed. Opt. Express 7, 2692 (2016).
[Crossref]

J. Märk, F.-J. Schmitt, and J. Laufer, J. Opt. 18, 054009 (2016).
[Crossref]

A. Forbrich, P. Shao, W. Shi, and R. J. Zemp, J. Opt. 18, 124001 (2016).
[Crossref]

2015 (1)

2014 (1)

J. Yao and L. V. Wang, Photoacoustics 2, 87 (2014).
[Crossref]

2013 (1)

A. M. Winkler, K. Maslov, and L. V. Wang, J. Biomed. Opt. 18, 097003 (2013).
[Crossref]

2012 (1)

C. Würth, M. G. González, R. Niessner, U. Panne, C. Haisch, and U. R. Genger, Talanta 90, 30 (2012).
[Crossref]

2011 (1)

A. M. Brouwer, Pure Appl. Chem. 83, 2213 (2011).
[Crossref]

2010 (1)

2009 (1)

K. Suzuki, A. Kobayashi, S. Kaneko, K. Takehira, T. Yoshihara, H. Ishida, Y. Shiina, S. Oishi, and S. Tobita, Phys. Chem. Chem. Phys. 11, 9850 (2009).
[Crossref]

2008 (1)

L. V. Wang, IEEE J. Sel. Top. Quantum Electron. 14, 171 (2008).
[Crossref]

2001 (1)

I. G. Calasso, W. Craig, and G. J. Diebold, Phys. Rev. Lett. 86, 3550 (2001).
[Crossref]

1999 (3)

C. G. A. Hoelen and F. F. M. de Mul, J. Acoust. Soc. Am. 106, 695 (1999).
[Crossref]

P. Harms, J. Sipior, N. Ram, G. M. Carter, and G. Rao, Rev. Sci. Instrum. 70, 1535 (1999).
[Crossref]

S. Fery-Forgues and D. Lavabre, J. Chem. Educ. 76, 1260 (1999).
[Crossref]

1992 (1)

J. R. Lakowicz, H. Szmacinski, K. Nowaczyk, K. W. Berndt, and M. Johnson, Anal. Biochem. 202, 316 (1992).
[Crossref]

1983 (2)

L. J. Rothberg, M. Bernstein, and K. S. Peters, J. Chem. Phys. 79, 2569 (1983).
[Crossref]

W. Keller, W. Schubert, R. Germer, and E. Strauss, J. Phys. Colloq. 44, C6-397 (1983).
[Crossref]

1982 (1)

D. Reiser and A. Laubereau, Appl. Phys. B 27, 115 (1982).
[Crossref]

1979 (1)

J. Olmsted, J. Phys. Chem. 83, 2581 (1979).
[Crossref]

1950 (1)

M. Kasha, Discuss. Faraday Soc. 9, 14 (1950).
[Crossref]

1880 (1)

A. G. Bell, Am. J. Sci. s3-20, 305 (1880).
[Crossref]

Bell, A. G.

A. G. Bell, Am. J. Sci. s3-20, 305 (1880).
[Crossref]

Berer, T.

Berndt, K. W.

J. R. Lakowicz, H. Szmacinski, K. Nowaczyk, K. W. Berndt, and M. Johnson, Anal. Biochem. 202, 316 (1992).
[Crossref]

Bernstein, M.

L. J. Rothberg, M. Bernstein, and K. S. Peters, J. Chem. Phys. 79, 2569 (1983).
[Crossref]

Brouwer, A. M.

A. M. Brouwer, Pure Appl. Chem. 83, 2213 (2011).
[Crossref]

Buchegger, B.

Calasso, I. G.

I. G. Calasso, W. Craig, and G. J. Diebold, Phys. Rev. Lett. 86, 3550 (2001).
[Crossref]

Carter, G. M.

P. Harms, J. Sipior, N. Ram, G. M. Carter, and G. Rao, Rev. Sci. Instrum. 70, 1535 (1999).
[Crossref]

Craig, W.

I. G. Calasso, W. Craig, and G. J. Diebold, Phys. Rev. Lett. 86, 3550 (2001).
[Crossref]

de Mul, F. F. M.

C. G. A. Hoelen and F. F. M. de Mul, J. Acoust. Soc. Am. 106, 695 (1999).
[Crossref]

Diebold, G. J.

I. G. Calasso, W. Craig, and G. J. Diebold, Phys. Rev. Lett. 86, 3550 (2001).
[Crossref]

Dortay, H.

Fery-Forgues, S.

S. Fery-Forgues and D. Lavabre, J. Chem. Educ. 76, 1260 (1999).
[Crossref]

Forbrich, A.

A. Forbrich, P. Shao, W. Shi, and R. J. Zemp, J. Opt. 18, 124001 (2016).
[Crossref]

Friedrich, T.

Genger, U. R.

C. Würth, M. G. González, R. Niessner, U. Panne, C. Haisch, and U. R. Genger, Talanta 90, 30 (2012).
[Crossref]

Germer, R.

W. Keller, W. Schubert, R. Germer, and E. Strauss, J. Phys. Colloq. 44, C6-397 (1983).
[Crossref]

González, M. G.

C. Würth, M. G. González, R. Niessner, U. Panne, C. Haisch, and U. R. Genger, Talanta 90, 30 (2012).
[Crossref]

Haisch, C.

C. Würth, M. G. González, R. Niessner, U. Panne, C. Haisch, and U. R. Genger, Talanta 90, 30 (2012).
[Crossref]

Harms, P.

P. Harms, J. Sipior, N. Ram, G. M. Carter, and G. Rao, Rev. Sci. Instrum. 70, 1535 (1999).
[Crossref]

Hoelen, C. G. A.

C. G. A. Hoelen and F. F. M. de Mul, J. Acoust. Soc. Am. 106, 695 (1999).
[Crossref]

Ishida, H.

K. Suzuki, A. Kobayashi, S. Kaneko, K. Takehira, T. Yoshihara, H. Ishida, Y. Shiina, S. Oishi, and S. Tobita, Phys. Chem. Chem. Phys. 11, 9850 (2009).
[Crossref]

Jacak, J.

Johnson, M.

J. R. Lakowicz, H. Szmacinski, K. Nowaczyk, K. W. Berndt, and M. Johnson, Anal. Biochem. 202, 316 (1992).
[Crossref]

Kaneko, S.

K. Suzuki, A. Kobayashi, S. Kaneko, K. Takehira, T. Yoshihara, H. Ishida, Y. Shiina, S. Oishi, and S. Tobita, Phys. Chem. Chem. Phys. 11, 9850 (2009).
[Crossref]

Kasha, M.

M. Kasha, Discuss. Faraday Soc. 9, 14 (1950).
[Crossref]

Keller, W.

W. Keller, W. Schubert, R. Germer, and E. Strauss, J. Phys. Colloq. 44, C6-397 (1983).
[Crossref]

Klar, T. A.

Kobayashi, A.

K. Suzuki, A. Kobayashi, S. Kaneko, K. Takehira, T. Yoshihara, H. Ishida, Y. Shiina, S. Oishi, and S. Tobita, Phys. Chem. Chem. Phys. 11, 9850 (2009).
[Crossref]

Kolios, M. C.

E. M. Strohm, M. J. Moore, and M. C. Kolios, Photoacoustics 4, 36 (2016).
[Crossref]

Lakowicz, J. R.

J. R. Lakowicz, H. Szmacinski, K. Nowaczyk, K. W. Berndt, and M. Johnson, Anal. Biochem. 202, 316 (1992).
[Crossref]

Langer, G.

Laubereau, A.

D. Reiser and A. Laubereau, Appl. Phys. B 27, 115 (1982).
[Crossref]

P. Troeger, C.-H. Liu, and A. Laubereau, Time Resolved Vibrational Spectroscopy, Springer Proceedings in Physics (Springer, 1985), pp. 62–66.

Laufer, J.

Lavabre, D.

S. Fery-Forgues and D. Lavabre, J. Chem. Educ. 76, 1260 (1999).
[Crossref]

Liu, C.-H.

P. Troeger, C.-H. Liu, and A. Laubereau, Time Resolved Vibrational Spectroscopy, Springer Proceedings in Physics (Springer, 1985), pp. 62–66.

Märk, J.

Maslov, K.

A. M. Winkler, K. Maslov, and L. V. Wang, J. Biomed. Opt. 18, 097003 (2013).
[Crossref]

C. Zhang, K. Maslov, and L. V. Wang, Opt. Lett. 35, 3195 (2010).
[Crossref]

Moore, M. J.

E. M. Strohm, M. J. Moore, and M. C. Kolios, Photoacoustics 4, 36 (2016).
[Crossref]

Niessner, R.

C. Würth, M. G. González, R. Niessner, U. Panne, C. Haisch, and U. R. Genger, Talanta 90, 30 (2012).
[Crossref]

Nowaczyk, K.

J. R. Lakowicz, H. Szmacinski, K. Nowaczyk, K. W. Berndt, and M. Johnson, Anal. Biochem. 202, 316 (1992).
[Crossref]

Oishi, S.

K. Suzuki, A. Kobayashi, S. Kaneko, K. Takehira, T. Yoshihara, H. Ishida, Y. Shiina, S. Oishi, and S. Tobita, Phys. Chem. Chem. Phys. 11, 9850 (2009).
[Crossref]

Olmsted, J.

J. Olmsted, J. Phys. Chem. 83, 2581 (1979).
[Crossref]

Panne, U.

C. Würth, M. G. González, R. Niessner, U. Panne, C. Haisch, and U. R. Genger, Talanta 90, 30 (2012).
[Crossref]

Peters, K. S.

L. J. Rothberg, M. Bernstein, and K. S. Peters, J. Chem. Phys. 79, 2569 (1983).
[Crossref]

Ram, N.

P. Harms, J. Sipior, N. Ram, G. M. Carter, and G. Rao, Rev. Sci. Instrum. 70, 1535 (1999).
[Crossref]

Rao, G.

P. Harms, J. Sipior, N. Ram, G. M. Carter, and G. Rao, Rev. Sci. Instrum. 70, 1535 (1999).
[Crossref]

Reiser, D.

D. Reiser and A. Laubereau, Appl. Phys. B 27, 115 (1982).
[Crossref]

Rothberg, L. J.

L. J. Rothberg, M. Bernstein, and K. S. Peters, J. Chem. Phys. 79, 2569 (1983).
[Crossref]

Schmitt, F.-J.

Schubert, W.

W. Keller, W. Schubert, R. Germer, and E. Strauss, J. Phys. Colloq. 44, C6-397 (1983).
[Crossref]

Shao, P.

A. Forbrich, P. Shao, W. Shi, and R. J. Zemp, J. Opt. 18, 124001 (2016).
[Crossref]

Shi, W.

A. Forbrich, P. Shao, W. Shi, and R. J. Zemp, J. Opt. 18, 124001 (2016).
[Crossref]

Shiina, Y.

K. Suzuki, A. Kobayashi, S. Kaneko, K. Takehira, T. Yoshihara, H. Ishida, Y. Shiina, S. Oishi, and S. Tobita, Phys. Chem. Chem. Phys. 11, 9850 (2009).
[Crossref]

Sipior, J.

P. Harms, J. Sipior, N. Ram, G. M. Carter, and G. Rao, Rev. Sci. Instrum. 70, 1535 (1999).
[Crossref]

Strauss, E.

W. Keller, W. Schubert, R. Germer, and E. Strauss, J. Phys. Colloq. 44, C6-397 (1983).
[Crossref]

Strohm, E. M.

E. M. Strohm, M. J. Moore, and M. C. Kolios, Photoacoustics 4, 36 (2016).
[Crossref]

Suzuki, K.

K. Suzuki, A. Kobayashi, S. Kaneko, K. Takehira, T. Yoshihara, H. Ishida, Y. Shiina, S. Oishi, and S. Tobita, Phys. Chem. Chem. Phys. 11, 9850 (2009).
[Crossref]

Szmacinski, H.

J. R. Lakowicz, H. Szmacinski, K. Nowaczyk, K. W. Berndt, and M. Johnson, Anal. Biochem. 202, 316 (1992).
[Crossref]

Takehira, K.

K. Suzuki, A. Kobayashi, S. Kaneko, K. Takehira, T. Yoshihara, H. Ishida, Y. Shiina, S. Oishi, and S. Tobita, Phys. Chem. Chem. Phys. 11, 9850 (2009).
[Crossref]

Theiss, C.

Tobita, S.

K. Suzuki, A. Kobayashi, S. Kaneko, K. Takehira, T. Yoshihara, H. Ishida, Y. Shiina, S. Oishi, and S. Tobita, Phys. Chem. Chem. Phys. 11, 9850 (2009).
[Crossref]

Troeger, P.

P. Troeger, C.-H. Liu, and A. Laubereau, Time Resolved Vibrational Spectroscopy, Springer Proceedings in Physics (Springer, 1985), pp. 62–66.

Wang, L. V.

J. Yao and L. V. Wang, Photoacoustics 2, 87 (2014).
[Crossref]

A. M. Winkler, K. Maslov, and L. V. Wang, J. Biomed. Opt. 18, 097003 (2013).
[Crossref]

C. Zhang, K. Maslov, and L. V. Wang, Opt. Lett. 35, 3195 (2010).
[Crossref]

L. V. Wang, IEEE J. Sel. Top. Quantum Electron. 14, 171 (2008).
[Crossref]

Winkler, A. M.

A. M. Winkler, K. Maslov, and L. V. Wang, J. Biomed. Opt. 18, 097003 (2013).
[Crossref]

Würth, C.

C. Würth, M. G. González, R. Niessner, U. Panne, C. Haisch, and U. R. Genger, Talanta 90, 30 (2012).
[Crossref]

Yao, J.

J. Yao and L. V. Wang, Photoacoustics 2, 87 (2014).
[Crossref]

Yoshihara, T.

K. Suzuki, A. Kobayashi, S. Kaneko, K. Takehira, T. Yoshihara, H. Ishida, Y. Shiina, S. Oishi, and S. Tobita, Phys. Chem. Chem. Phys. 11, 9850 (2009).
[Crossref]

Zemp, R. J.

A. Forbrich, P. Shao, W. Shi, and R. J. Zemp, J. Opt. 18, 124001 (2016).
[Crossref]

Zhang, C.

Am. J. Sci. (1)

A. G. Bell, Am. J. Sci. s3-20, 305 (1880).
[Crossref]

Anal. Biochem. (1)

J. R. Lakowicz, H. Szmacinski, K. Nowaczyk, K. W. Berndt, and M. Johnson, Anal. Biochem. 202, 316 (1992).
[Crossref]

Appl. Phys. B (1)

D. Reiser and A. Laubereau, Appl. Phys. B 27, 115 (1982).
[Crossref]

Biomed. Opt. Express (2)

Discuss. Faraday Soc. (1)

M. Kasha, Discuss. Faraday Soc. 9, 14 (1950).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

L. V. Wang, IEEE J. Sel. Top. Quantum Electron. 14, 171 (2008).
[Crossref]

J. Acoust. Soc. Am. (1)

C. G. A. Hoelen and F. F. M. de Mul, J. Acoust. Soc. Am. 106, 695 (1999).
[Crossref]

J. Biomed. Opt. (1)

A. M. Winkler, K. Maslov, and L. V. Wang, J. Biomed. Opt. 18, 097003 (2013).
[Crossref]

J. Chem. Educ. (1)

S. Fery-Forgues and D. Lavabre, J. Chem. Educ. 76, 1260 (1999).
[Crossref]

J. Chem. Phys. (1)

L. J. Rothberg, M. Bernstein, and K. S. Peters, J. Chem. Phys. 79, 2569 (1983).
[Crossref]

J. Opt. (2)

J. Märk, F.-J. Schmitt, and J. Laufer, J. Opt. 18, 054009 (2016).
[Crossref]

A. Forbrich, P. Shao, W. Shi, and R. J. Zemp, J. Opt. 18, 124001 (2016).
[Crossref]

J. Phys. Chem. (1)

J. Olmsted, J. Phys. Chem. 83, 2581 (1979).
[Crossref]

J. Phys. Colloq. (1)

W. Keller, W. Schubert, R. Germer, and E. Strauss, J. Phys. Colloq. 44, C6-397 (1983).
[Crossref]

Opt. Lett. (1)

Photoacoustics (2)

E. M. Strohm, M. J. Moore, and M. C. Kolios, Photoacoustics 4, 36 (2016).
[Crossref]

J. Yao and L. V. Wang, Photoacoustics 2, 87 (2014).
[Crossref]

Phys. Chem. Chem. Phys. (1)

K. Suzuki, A. Kobayashi, S. Kaneko, K. Takehira, T. Yoshihara, H. Ishida, Y. Shiina, S. Oishi, and S. Tobita, Phys. Chem. Chem. Phys. 11, 9850 (2009).
[Crossref]

Phys. Rev. Lett. (1)

I. G. Calasso, W. Craig, and G. J. Diebold, Phys. Rev. Lett. 86, 3550 (2001).
[Crossref]

Proc. SPIE (1)

G. Langer and T. Berer, Proc. SPIE 10494, 1049465 (2018).
[Crossref]

Pure Appl. Chem. (1)

A. M. Brouwer, Pure Appl. Chem. 83, 2213 (2011).
[Crossref]

Rev. Sci. Instrum. (1)

P. Harms, J. Sipior, N. Ram, G. M. Carter, and G. Rao, Rev. Sci. Instrum. 70, 1535 (1999).
[Crossref]

Talanta (1)

C. Würth, M. G. González, R. Niessner, U. Panne, C. Haisch, and U. R. Genger, Talanta 90, 30 (2012).
[Crossref]

Other (2)

ATTO-TEC GmbH, “ATTO-TEC fluorescent labels and dyes product catalogue 2016/2018” (2016).

P. Troeger, C.-H. Liu, and A. Laubereau, Time Resolved Vibrational Spectroscopy, Springer Proceedings in Physics (Springer, 1985), pp. 62–66.

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Figures (3)

Fig. 1.
Fig. 1. First, a photon with an energy E exc = E v 0 + E fl + E v 1 excites an electron from the ground state S 0 into an excited state S 1 * (exc, blue). Second, the electron releases the energy E v 1 within several picoseconds ( S 1 * S 1 , red wavy path). Third, the electron returns into the HOMO either radiatively (fl, green) or nonradiatively (th, red). Finally, the electron returns to the ground state S 0 (red wavy path).
Fig. 2.
Fig. 2. (a) Dissipated energy amplitude E as a function of frequency f for the different relaxation paths. The curves are normalized to their respective released energies. The fluorescence signal (Fl, bold green) and heat released via relaxation from S 1 ( H fl , v 0 , red) behave similarly, as both share the same time constant τ . The overall heating function (bold black) does not go to zero for high frequencies, because of the S 1 * S 1 vibrational relaxation contribution ( H v 1 , blue). (b) The PA pressure amplitude p is proportional to the time derivative of the heating function. The contribution of the S 1 * S 1 relaxation ( p v 1 , blue) increases linearly with frequency; relaxation from S 1 (red, p fl , v 0 ) shows a high-pass-like behavior. The bold black line illustrates the overall generated pressure.
Fig. 3.
Fig. 3. (a) PA pressure amplitude p as a function of frequency [Eq. (6)] for different fluorescence quantum efficiencies η and τ = 5    ns . The corresponding PA phase response according to Eq. (7) is shown in (b). (c) PA pressure amplitude for different excited state lifetimes τ and η = 0.9 . The dashed line shows the contribution of the vibrational relaxation within the excited state. (d) PA phase ϕ corresponding to (c).

Equations (11)

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

E fl = E f ( E ) d E f ( E ) d E ,
f ( t ) = N S 1 , 0 η τ e t / τ Θ ( t ) ,
f ( ω ) = N S 1 , 0 η 2 π 1 1 + i ω τ ,
h ( t ) = N S 1 , 0 δ ( t ) E v 1 + N S 1 , 0 τ e t / τ E fl , v 0 ,
p ( ω ) ( i ω E v 1 + i ω 1 + i ω τ E fl , v 0 ) N S 1 , 0 2 π .
| p ( ω ) | = ( ω 1 + τ 2 ω 2 E fl , v 0 + ω E v 1 ) N S 1 , 0 2 π ,
ϕ ( ω ) = arctan ( E v 1 + E fl , v 0 + E v 1 · τ 2 ω 2 E fl , v 0 · τ ω ) .
ϕ min = arctan ( 2 · E v 1 + E fl , v 0 E v 1 E fl , v 0 ) ω min = E v 1 + E fl , v 0 τ 2 · E v 1 = E v 1 + ( 1 η ) E fl + E v 0 τ 2 · E v 1 .
E S 1 = 2 1 + tan 2 ( ϕ min ) 1 tan 2 ( ϕ min ) · E exc .
τ = E exc ( 2 π f min ) 2 ( E exc E S 1 ) ,
η = E exc ( 2 π f min τ ) 2 ( E exc E S 1 ) E fl .

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