Abstract

Time-delay of transmitted pulses with respect to the incident pulse in bacteriorhodopsin films has been studied without the use of a pump beam. Based on a modified saturable absorber model, analytical expressions of the transmitted pulse have been obtained. As a result, time delay, distortion and fractional delay have been analyzed for sinusoidal pulses with a low background. A good agreement between theory and experiences has been observed.

© 2014 Optical Society of America

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References

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  1. P. F. Wu and D. V. G. L. N. Rao, “Controllable snail-paced light in biological bacteriorhodopsin thin film,” Phys. Rev. Lett.95(2005).
    [Crossref] [PubMed]
  2. C. S. Yelleswarapu, R. Philip, F. J. Aranda, B. R. Kimball, and D. V. G. L. N. Rao, “Slow light in bacteriorhodopsin solution using coherent population oscillations,” Opt. Lett. 32, 1788–1790 (2007).
    [Crossref] [PubMed]
  3. C. S. Yelleswarapu, S. Laoui, R. Philip, and D. V. G. L. N. Rao, “Coherent population oscillations and superluminal light in a protein complex,” Opt. Express 16, 3844–3852 (2008).
    [Crossref] [PubMed]
  4. V. S. Zapasskii and G. G. Kozlov, “A saturable absorber, coherent population oscillations, and slow light,” Opt. Spectrosc. 100, 419–424 (2006).
    [Crossref]
  5. B. Macke and B. Segard, “Slow light in saturable absorbers,” Phys. Rev. A78(2008).
    [Crossref]
  6. A. C. Selden, “Slow light and saturable absorption,” Opt. Spectrosc. 106, 881–888 (2009).
    [Crossref]
  7. A. C. Selden, “Practical tests for distinguishing slow light from saturable absorption,” Opt. Express 18, 13204–13211 (2010).
    [Crossref] [PubMed]
  8. F. Gires and F. Combaud, “Saturation de l’absorption optique de certaines solutions de phtalocyanines,” J. Phys. (Paris) 26, 325–330 (1965).
    [Crossref]
  9. A. C. Selden, “Pulse transmission through a saturable absorber,” Brit. J. Appl. Phys. 18, 743–748 (1967).
    [Crossref]
  10. P. Acebal, S. Blaya, L. Carretero, R. F. Madrigal, and A. Fimia, “Rigorous analysis of the propagation of sinusoidal pulses in bacteriorhodopsin films,” Opt. Express 20, 25497–25512 (2012).
    [Crossref] [PubMed]
  11. N. Hampp, A. Popp, C. Bruchle, and D. Oesterhelt, “Diffraction efficiency of bacteriorhodopsin films for holography containing bacteriorhodopsin wildtype BRWT and its variants BRD85E and BRD96N,” J. Phys. Chem. 96(11) 4679–4685 (1992).
    [Crossref]
  12. J. D. Downie and D. T. Smithey, “Measurements of holographic properties of bacteriorhodopsin films,” Appl. Opt. 35, 5780–5789 (1996).
    [Crossref] [PubMed]
  13. O. Werner, B. Fischer, A. Lewis, and I. Nebenzahl, “Saturable absorption, wave mixing, and phase conjugation with bacteriorhodopsin,” Opt. Lett. 15, 1117–1119 (1990).
    [Crossref] [PubMed]
  14. J. D. Downie and D. A. Timucin, “Modeling the grating-formation process in thick bacteriorhodopsin films,” Appl. Opt. 37, 2102–2111 (1998).
    [Crossref]
  15. B. Macke and B. Segard, “Propagation of light-pulses at a negative group-velocity,” Eur. Phys. J. D 23, 125–141 (2003).
    [Crossref]
  16. M. S. Bigelow, N. N. Lepeshkin, H. Shin, and R. W. Boyd, “Propagation of smooth and discontinuous pulses through materials with very large or very small group velocities,” J. Phys. Condens. Matter 18, 3117–3126 (2006).
    [Crossref]
  17. P. Acebal, L. Carretero, S. Blaya, A. Murciano, and A. Fimia, “Theoretical approach to photoinduced inhomogeneous anisotropy in bacteriorhodopsin films,” Phys. Rev. E 76, 016608 (2007).
    [Crossref]

2012 (1)

P. Acebal, S. Blaya, L. Carretero, R. F. Madrigal, and A. Fimia, “Rigorous analysis of the propagation of sinusoidal pulses in bacteriorhodopsin films,” Opt. Express 20, 25497–25512 (2012).
[Crossref] [PubMed]

2010 (1)

A. C. Selden, “Practical tests for distinguishing slow light from saturable absorption,” Opt. Express 18, 13204–13211 (2010).
[Crossref] [PubMed]

2009 (1)

A. C. Selden, “Slow light and saturable absorption,” Opt. Spectrosc. 106, 881–888 (2009).
[Crossref]

2008 (1)

C. S. Yelleswarapu, S. Laoui, R. Philip, and D. V. G. L. N. Rao, “Coherent population oscillations and superluminal light in a protein complex,” Opt. Express 16, 3844–3852 (2008).
[Crossref] [PubMed]

2007 (2)

C. S. Yelleswarapu, R. Philip, F. J. Aranda, B. R. Kimball, and D. V. G. L. N. Rao, “Slow light in bacteriorhodopsin solution using coherent population oscillations,” Opt. Lett. 32, 1788–1790 (2007).
[Crossref] [PubMed]

P. Acebal, L. Carretero, S. Blaya, A. Murciano, and A. Fimia, “Theoretical approach to photoinduced inhomogeneous anisotropy in bacteriorhodopsin films,” Phys. Rev. E 76, 016608 (2007).
[Crossref]

2006 (2)

M. S. Bigelow, N. N. Lepeshkin, H. Shin, and R. W. Boyd, “Propagation of smooth and discontinuous pulses through materials with very large or very small group velocities,” J. Phys. Condens. Matter 18, 3117–3126 (2006).
[Crossref]

V. S. Zapasskii and G. G. Kozlov, “A saturable absorber, coherent population oscillations, and slow light,” Opt. Spectrosc. 100, 419–424 (2006).
[Crossref]

2003 (1)

B. Macke and B. Segard, “Propagation of light-pulses at a negative group-velocity,” Eur. Phys. J. D 23, 125–141 (2003).
[Crossref]

1998 (1)

J. D. Downie and D. A. Timucin, “Modeling the grating-formation process in thick bacteriorhodopsin films,” Appl. Opt. 37, 2102–2111 (1998).
[Crossref]

1996 (1)

J. D. Downie and D. T. Smithey, “Measurements of holographic properties of bacteriorhodopsin films,” Appl. Opt. 35, 5780–5789 (1996).
[Crossref] [PubMed]

1992 (1)

N. Hampp, A. Popp, C. Bruchle, and D. Oesterhelt, “Diffraction efficiency of bacteriorhodopsin films for holography containing bacteriorhodopsin wildtype BRWT and its variants BRD85E and BRD96N,” J. Phys. Chem. 96(11) 4679–4685 (1992).
[Crossref]

1990 (1)

O. Werner, B. Fischer, A. Lewis, and I. Nebenzahl, “Saturable absorption, wave mixing, and phase conjugation with bacteriorhodopsin,” Opt. Lett. 15, 1117–1119 (1990).
[Crossref] [PubMed]

1967 (1)

A. C. Selden, “Pulse transmission through a saturable absorber,” Brit. J. Appl. Phys. 18, 743–748 (1967).
[Crossref]

1965 (1)

F. Gires and F. Combaud, “Saturation de l’absorption optique de certaines solutions de phtalocyanines,” J. Phys. (Paris) 26, 325–330 (1965).
[Crossref]

Acebal, P.

P. Acebal, S. Blaya, L. Carretero, R. F. Madrigal, and A. Fimia, “Rigorous analysis of the propagation of sinusoidal pulses in bacteriorhodopsin films,” Opt. Express 20, 25497–25512 (2012).
[Crossref] [PubMed]

P. Acebal, L. Carretero, S. Blaya, A. Murciano, and A. Fimia, “Theoretical approach to photoinduced inhomogeneous anisotropy in bacteriorhodopsin films,” Phys. Rev. E 76, 016608 (2007).
[Crossref]

Aranda, F. J.

C. S. Yelleswarapu, R. Philip, F. J. Aranda, B. R. Kimball, and D. V. G. L. N. Rao, “Slow light in bacteriorhodopsin solution using coherent population oscillations,” Opt. Lett. 32, 1788–1790 (2007).
[Crossref] [PubMed]

Bigelow, M. S.

M. S. Bigelow, N. N. Lepeshkin, H. Shin, and R. W. Boyd, “Propagation of smooth and discontinuous pulses through materials with very large or very small group velocities,” J. Phys. Condens. Matter 18, 3117–3126 (2006).
[Crossref]

Blaya, S.

P. Acebal, S. Blaya, L. Carretero, R. F. Madrigal, and A. Fimia, “Rigorous analysis of the propagation of sinusoidal pulses in bacteriorhodopsin films,” Opt. Express 20, 25497–25512 (2012).
[Crossref] [PubMed]

P. Acebal, L. Carretero, S. Blaya, A. Murciano, and A. Fimia, “Theoretical approach to photoinduced inhomogeneous anisotropy in bacteriorhodopsin films,” Phys. Rev. E 76, 016608 (2007).
[Crossref]

Boyd, R. W.

M. S. Bigelow, N. N. Lepeshkin, H. Shin, and R. W. Boyd, “Propagation of smooth and discontinuous pulses through materials with very large or very small group velocities,” J. Phys. Condens. Matter 18, 3117–3126 (2006).
[Crossref]

Bruchle, C.

N. Hampp, A. Popp, C. Bruchle, and D. Oesterhelt, “Diffraction efficiency of bacteriorhodopsin films for holography containing bacteriorhodopsin wildtype BRWT and its variants BRD85E and BRD96N,” J. Phys. Chem. 96(11) 4679–4685 (1992).
[Crossref]

Carretero, L.

P. Acebal, S. Blaya, L. Carretero, R. F. Madrigal, and A. Fimia, “Rigorous analysis of the propagation of sinusoidal pulses in bacteriorhodopsin films,” Opt. Express 20, 25497–25512 (2012).
[Crossref] [PubMed]

P. Acebal, L. Carretero, S. Blaya, A. Murciano, and A. Fimia, “Theoretical approach to photoinduced inhomogeneous anisotropy in bacteriorhodopsin films,” Phys. Rev. E 76, 016608 (2007).
[Crossref]

Combaud, F.

F. Gires and F. Combaud, “Saturation de l’absorption optique de certaines solutions de phtalocyanines,” J. Phys. (Paris) 26, 325–330 (1965).
[Crossref]

Downie, J. D.

J. D. Downie and D. A. Timucin, “Modeling the grating-formation process in thick bacteriorhodopsin films,” Appl. Opt. 37, 2102–2111 (1998).
[Crossref]

J. D. Downie and D. T. Smithey, “Measurements of holographic properties of bacteriorhodopsin films,” Appl. Opt. 35, 5780–5789 (1996).
[Crossref] [PubMed]

Fimia, A.

P. Acebal, S. Blaya, L. Carretero, R. F. Madrigal, and A. Fimia, “Rigorous analysis of the propagation of sinusoidal pulses in bacteriorhodopsin films,” Opt. Express 20, 25497–25512 (2012).
[Crossref] [PubMed]

P. Acebal, L. Carretero, S. Blaya, A. Murciano, and A. Fimia, “Theoretical approach to photoinduced inhomogeneous anisotropy in bacteriorhodopsin films,” Phys. Rev. E 76, 016608 (2007).
[Crossref]

Fischer, B.

O. Werner, B. Fischer, A. Lewis, and I. Nebenzahl, “Saturable absorption, wave mixing, and phase conjugation with bacteriorhodopsin,” Opt. Lett. 15, 1117–1119 (1990).
[Crossref] [PubMed]

Gires, F.

F. Gires and F. Combaud, “Saturation de l’absorption optique de certaines solutions de phtalocyanines,” J. Phys. (Paris) 26, 325–330 (1965).
[Crossref]

Hampp, N.

N. Hampp, A. Popp, C. Bruchle, and D. Oesterhelt, “Diffraction efficiency of bacteriorhodopsin films for holography containing bacteriorhodopsin wildtype BRWT and its variants BRD85E and BRD96N,” J. Phys. Chem. 96(11) 4679–4685 (1992).
[Crossref]

Kimball, B. R.

C. S. Yelleswarapu, R. Philip, F. J. Aranda, B. R. Kimball, and D. V. G. L. N. Rao, “Slow light in bacteriorhodopsin solution using coherent population oscillations,” Opt. Lett. 32, 1788–1790 (2007).
[Crossref] [PubMed]

Kozlov, G. G.

V. S. Zapasskii and G. G. Kozlov, “A saturable absorber, coherent population oscillations, and slow light,” Opt. Spectrosc. 100, 419–424 (2006).
[Crossref]

Laoui, S.

C. S. Yelleswarapu, S. Laoui, R. Philip, and D. V. G. L. N. Rao, “Coherent population oscillations and superluminal light in a protein complex,” Opt. Express 16, 3844–3852 (2008).
[Crossref] [PubMed]

Lepeshkin, N. N.

M. S. Bigelow, N. N. Lepeshkin, H. Shin, and R. W. Boyd, “Propagation of smooth and discontinuous pulses through materials with very large or very small group velocities,” J. Phys. Condens. Matter 18, 3117–3126 (2006).
[Crossref]

Lewis, A.

O. Werner, B. Fischer, A. Lewis, and I. Nebenzahl, “Saturable absorption, wave mixing, and phase conjugation with bacteriorhodopsin,” Opt. Lett. 15, 1117–1119 (1990).
[Crossref] [PubMed]

Macke, B.

B. Macke and B. Segard, “Propagation of light-pulses at a negative group-velocity,” Eur. Phys. J. D 23, 125–141 (2003).
[Crossref]

B. Macke and B. Segard, “Slow light in saturable absorbers,” Phys. Rev. A78(2008).
[Crossref]

Madrigal, R. F.

P. Acebal, S. Blaya, L. Carretero, R. F. Madrigal, and A. Fimia, “Rigorous analysis of the propagation of sinusoidal pulses in bacteriorhodopsin films,” Opt. Express 20, 25497–25512 (2012).
[Crossref] [PubMed]

Murciano, A.

P. Acebal, L. Carretero, S. Blaya, A. Murciano, and A. Fimia, “Theoretical approach to photoinduced inhomogeneous anisotropy in bacteriorhodopsin films,” Phys. Rev. E 76, 016608 (2007).
[Crossref]

Nebenzahl, I.

O. Werner, B. Fischer, A. Lewis, and I. Nebenzahl, “Saturable absorption, wave mixing, and phase conjugation with bacteriorhodopsin,” Opt. Lett. 15, 1117–1119 (1990).
[Crossref] [PubMed]

Oesterhelt, D.

N. Hampp, A. Popp, C. Bruchle, and D. Oesterhelt, “Diffraction efficiency of bacteriorhodopsin films for holography containing bacteriorhodopsin wildtype BRWT and its variants BRD85E and BRD96N,” J. Phys. Chem. 96(11) 4679–4685 (1992).
[Crossref]

Philip, R.

C. S. Yelleswarapu, S. Laoui, R. Philip, and D. V. G. L. N. Rao, “Coherent population oscillations and superluminal light in a protein complex,” Opt. Express 16, 3844–3852 (2008).
[Crossref] [PubMed]

C. S. Yelleswarapu, R. Philip, F. J. Aranda, B. R. Kimball, and D. V. G. L. N. Rao, “Slow light in bacteriorhodopsin solution using coherent population oscillations,” Opt. Lett. 32, 1788–1790 (2007).
[Crossref] [PubMed]

Popp, A.

N. Hampp, A. Popp, C. Bruchle, and D. Oesterhelt, “Diffraction efficiency of bacteriorhodopsin films for holography containing bacteriorhodopsin wildtype BRWT and its variants BRD85E and BRD96N,” J. Phys. Chem. 96(11) 4679–4685 (1992).
[Crossref]

Rao, D. V. G. L. N.

C. S. Yelleswarapu, S. Laoui, R. Philip, and D. V. G. L. N. Rao, “Coherent population oscillations and superluminal light in a protein complex,” Opt. Express 16, 3844–3852 (2008).
[Crossref] [PubMed]

C. S. Yelleswarapu, R. Philip, F. J. Aranda, B. R. Kimball, and D. V. G. L. N. Rao, “Slow light in bacteriorhodopsin solution using coherent population oscillations,” Opt. Lett. 32, 1788–1790 (2007).
[Crossref] [PubMed]

P. F. Wu and D. V. G. L. N. Rao, “Controllable snail-paced light in biological bacteriorhodopsin thin film,” Phys. Rev. Lett.95(2005).
[Crossref] [PubMed]

Segard, B.

B. Macke and B. Segard, “Propagation of light-pulses at a negative group-velocity,” Eur. Phys. J. D 23, 125–141 (2003).
[Crossref]

B. Macke and B. Segard, “Slow light in saturable absorbers,” Phys. Rev. A78(2008).
[Crossref]

Selden, A. C.

A. C. Selden, “Practical tests for distinguishing slow light from saturable absorption,” Opt. Express 18, 13204–13211 (2010).
[Crossref] [PubMed]

A. C. Selden, “Slow light and saturable absorption,” Opt. Spectrosc. 106, 881–888 (2009).
[Crossref]

A. C. Selden, “Pulse transmission through a saturable absorber,” Brit. J. Appl. Phys. 18, 743–748 (1967).
[Crossref]

Shin, H.

M. S. Bigelow, N. N. Lepeshkin, H. Shin, and R. W. Boyd, “Propagation of smooth and discontinuous pulses through materials with very large or very small group velocities,” J. Phys. Condens. Matter 18, 3117–3126 (2006).
[Crossref]

Smithey, D. T.

J. D. Downie and D. T. Smithey, “Measurements of holographic properties of bacteriorhodopsin films,” Appl. Opt. 35, 5780–5789 (1996).
[Crossref] [PubMed]

Timucin, D. A.

J. D. Downie and D. A. Timucin, “Modeling the grating-formation process in thick bacteriorhodopsin films,” Appl. Opt. 37, 2102–2111 (1998).
[Crossref]

Werner, O.

O. Werner, B. Fischer, A. Lewis, and I. Nebenzahl, “Saturable absorption, wave mixing, and phase conjugation with bacteriorhodopsin,” Opt. Lett. 15, 1117–1119 (1990).
[Crossref] [PubMed]

Wu, P. F.

P. F. Wu and D. V. G. L. N. Rao, “Controllable snail-paced light in biological bacteriorhodopsin thin film,” Phys. Rev. Lett.95(2005).
[Crossref] [PubMed]

Yelleswarapu, C. S.

C. S. Yelleswarapu, S. Laoui, R. Philip, and D. V. G. L. N. Rao, “Coherent population oscillations and superluminal light in a protein complex,” Opt. Express 16, 3844–3852 (2008).
[Crossref] [PubMed]

C. S. Yelleswarapu, R. Philip, F. J. Aranda, B. R. Kimball, and D. V. G. L. N. Rao, “Slow light in bacteriorhodopsin solution using coherent population oscillations,” Opt. Lett. 32, 1788–1790 (2007).
[Crossref] [PubMed]

Zapasskii, V. S.

V. S. Zapasskii and G. G. Kozlov, “A saturable absorber, coherent population oscillations, and slow light,” Opt. Spectrosc. 100, 419–424 (2006).
[Crossref]

Appl. Opt. (2)

J. D. Downie and D. T. Smithey, “Measurements of holographic properties of bacteriorhodopsin films,” Appl. Opt. 35, 5780–5789 (1996).
[Crossref] [PubMed]

J. D. Downie and D. A. Timucin, “Modeling the grating-formation process in thick bacteriorhodopsin films,” Appl. Opt. 37, 2102–2111 (1998).
[Crossref]

Brit. J. Appl. Phys. (1)

A. C. Selden, “Pulse transmission through a saturable absorber,” Brit. J. Appl. Phys. 18, 743–748 (1967).
[Crossref]

Eur. Phys. J. D (1)

B. Macke and B. Segard, “Propagation of light-pulses at a negative group-velocity,” Eur. Phys. J. D 23, 125–141 (2003).
[Crossref]

J. Phys. (Paris) (1)

F. Gires and F. Combaud, “Saturation de l’absorption optique de certaines solutions de phtalocyanines,” J. Phys. (Paris) 26, 325–330 (1965).
[Crossref]

J. Phys. Chem. (1)

N. Hampp, A. Popp, C. Bruchle, and D. Oesterhelt, “Diffraction efficiency of bacteriorhodopsin films for holography containing bacteriorhodopsin wildtype BRWT and its variants BRD85E and BRD96N,” J. Phys. Chem. 96(11) 4679–4685 (1992).
[Crossref]

J. Phys. Condens. Matter (1)

M. S. Bigelow, N. N. Lepeshkin, H. Shin, and R. W. Boyd, “Propagation of smooth and discontinuous pulses through materials with very large or very small group velocities,” J. Phys. Condens. Matter 18, 3117–3126 (2006).
[Crossref]

Opt. Express (3)

P. Acebal, S. Blaya, L. Carretero, R. F. Madrigal, and A. Fimia, “Rigorous analysis of the propagation of sinusoidal pulses in bacteriorhodopsin films,” Opt. Express 20, 25497–25512 (2012).
[Crossref] [PubMed]

A. C. Selden, “Practical tests for distinguishing slow light from saturable absorption,” Opt. Express 18, 13204–13211 (2010).
[Crossref] [PubMed]

C. S. Yelleswarapu, S. Laoui, R. Philip, and D. V. G. L. N. Rao, “Coherent population oscillations and superluminal light in a protein complex,” Opt. Express 16, 3844–3852 (2008).
[Crossref] [PubMed]

Opt. Lett. (2)

C. S. Yelleswarapu, R. Philip, F. J. Aranda, B. R. Kimball, and D. V. G. L. N. Rao, “Slow light in bacteriorhodopsin solution using coherent population oscillations,” Opt. Lett. 32, 1788–1790 (2007).
[Crossref] [PubMed]

O. Werner, B. Fischer, A. Lewis, and I. Nebenzahl, “Saturable absorption, wave mixing, and phase conjugation with bacteriorhodopsin,” Opt. Lett. 15, 1117–1119 (1990).
[Crossref] [PubMed]

Opt. Spectrosc. (2)

V. S. Zapasskii and G. G. Kozlov, “A saturable absorber, coherent population oscillations, and slow light,” Opt. Spectrosc. 100, 419–424 (2006).
[Crossref]

A. C. Selden, “Slow light and saturable absorption,” Opt. Spectrosc. 106, 881–888 (2009).
[Crossref]

Phys. Rev. E (1)

P. Acebal, L. Carretero, S. Blaya, A. Murciano, and A. Fimia, “Theoretical approach to photoinduced inhomogeneous anisotropy in bacteriorhodopsin films,” Phys. Rev. E 76, 016608 (2007).
[Crossref]

Other (2)

B. Macke and B. Segard, “Slow light in saturable absorbers,” Phys. Rev. A78(2008).
[Crossref]

P. F. Wu and D. V. G. L. N. Rao, “Controllable snail-paced light in biological bacteriorhodopsin thin film,” Phys. Rev. Lett.95(2005).
[Crossref] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

The experimental setup used to analyze the propagation of sinusoidal pulses in bacteriorhodopsin film.

Fig. 2
Fig. 2

Temporal variation of the experimental and fitted sinusoidal modulated beam (signal) and the corresponding experimental and fitted reference beam, where Cin = 0.4 mW/cm2, I0 = 5.5 mW/cm2 and the regression coefficients of the reference and signal beams were 0.999 and 0.992 respectively.

Fig. 3
Fig. 3

Variation of the fitted parameters of the signal intensity curves as a function of the total intensity: τM (a) and α0L (b). The ratio Cin/I0 oscillates between 0.05 to 0.12. Orange line correspond to the mean value of all the obtained parameters.

Fig. 4
Fig. 4

Variation of the time delay of a sinusoidal modulated beam (a), fractional delay (b) and distortion of the pulse as a function of the total intensity. The ratio Cin/I0 oscillates between 0.05 to 0.12. Theoretical simulations by using Eq. 18 taking into account Eq. 24, the mean value of α0L and τM are shown for each frequency (blue line 0.2 Hz, red line 0.7 Hz, green line 1 Hz and black line 1.5 Hz).

Equations (25)

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

M t = ϕ B σ B I B ϕ M σ M I M M τ M
I z = ( σ B I B + σ M I M )
N 0 = B + M
τ M N t = β 1 I N ( β 2 I + 1 )
τ M ϕ B N 0 I z = I ( β 3 N + β 1 )
z ( τ M ln I t + ln I + β 2 I + β 1 ϕ B N 0 τ M z ) = 0
τ M ln I out t + ln I out + β 2 I out + α 0 L = τ M ln I in t + ln I in + β 2 I in
τ M ln T t + ln T + β 2 I in ( T 1 ) + α 0 L = 0
ln T + β 2 I in ( T 1 ) + α 0 L = 0
τ M ln T t + ln T + α 0 L β 2 I in ( t )
I in = C in + S in ( t )
I out = C out + S out ( t )
Z ( t ) = ln T ( t ) + α 0 L β 2 C in
τ M Z ( t ) t + Z ( t ) = β 2 S in ( t )
Z ( t ) = β 2 e t τ M τ M t 0 t S in ( θ ) e θ τ M d θ
( C out + S out ( t ) ) = ( C in + S in ( t ) ) e Z ( t ) α 0 L + β 2 C in
C out = C in e α 0 L + β 2 C in
S out ( t ) = ( C in ( e Z ( t ) 1 ) + S in ( t ) e Z ( t ) ) e α 0 L + β 2 C in
d S out ( t ) d t | t smax = d Z ( t ) d t | t smax ( C in + S in ( t smax ) ) e Z ( t smax ) α 0 L + β 2 C in
d Z ( t ) d t | t smax = τ M 1 ( β 2 S in ( t smax ) Z ( t smax ) )
Z ( t smax ) = β 2 e t smax τ M τ M t 0 t smax S in ( θ ) e θ τ M d θ
Z ( t smax ) = β 2 e t smax τ M τ M t 0 t smax S in ( θ ) e θ τ M d θ β 2 e t smax τ M S in ( t smax ) ( e t smax τ M e t 0 τ M ) β 2 S in ( t smax ) ( 1 e ( t 0 + t smax ) τ M ) β 2 S in ( t smax )
S in ( t ) = I 0 sin 2 ( π ( t t 0 ) τ in )
Z ( t ) = β 2 I 0 ( τ in 2 + 4 ( 1 e ( t t 0 ) τ M ) π 2 τ M 2 τ in ( 2 π τ M sin ( 2 π ( t t 0 ) τ in ) + τ in cos ( 2 π ( t t 0 ) τ in ) ) ) 2 ( τ in 2 + 4 π 2 τ M 2 )
D = ( t 0 t 1 | Sn out ( t ) Sn in ( t τ D ) | d t t 0 t 1 Sn in ( t ) d t ) 1 2

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