Abstract

Fluorescence lifetime imaging microscopy is used widely in biological research, but the accuracy and precision of lifetime measurements are limited. Photon noise is an inherent error source that cannot be eliminated. In this paper, we present a general approach to compute the probability density of the estimated lifetime for frequency-domain fluorescence lifetime imaging microscopy using homodyne lock-in detection. The analysis for commonly used excitation methods, including sinusoidal modulation, square-wave modulation, and a periodically pulsed light source, are given and compared to the results of Monte Carlo simulations. The optimum parameters of the excitation waveforms to minimize the variance of the estimated lifetimes are also derived and compared to previously published results.

© 2010 Optical Society of America

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  1. K. M. Hanson, M. J. Behne, N. P. Barry, T. M. Mauro, E. Gratton, and R. M. Clegg, “Two-photon fluorescence lifetime imaging of the skin stratum corneum pH gradient,” Biophys. J. 83, 1682–1690 (2002).
    [CrossRef] [PubMed]
  2. H.-J. Lin, P. Herman, and J. R. Lakowicz, “Fluorescence lifetime-resolved pH imaging of living cells,” Cytometry, Part A 52A, 77–89 (2003).
    [CrossRef]
  3. M. C. Skala, K. M. Riching, A. Gendron-Fitzpatrick, J. Eickhoff, K. W. Eliceiri, J. G. White, and N. Ramanujam, “In vivo multiphoton microscopy of NADH and FAD redox states, fluorescence lifetimes, and cellular morphology in precancerous epithelia,” Proc. Natl. Acad. Sci. U.S.A. 104, 19494–19499 (2007).
    [CrossRef] [PubMed]
  4. T. Vo-Dinh, Biomedical Photonics Handbook (CRC, 2003).
    [CrossRef]
  5. R. Sanders, A. Draaijer, H. C. Gerritsen, P. M. Houpt, and Y. K. Levine, “Quantitative pH imaging in cells using confocal fluorescence lifetime imaging microscopy,” Anal. Biochem. 227, 302–308 (1995).
    [CrossRef] [PubMed]
  6. E. B. van Munster and T. W. J. Gadella, “Fluorescence lifetime imaging microscopy (FLIM),” Adv. Biochem. Eng./Biotechnol. 95, 143–175 (2005).
  7. A. Draaijer, R. Sanders, and H. C. Gerritsen, “Fluorescence lifetime imaging, a new tool in confocal microscopy” in Handbook of Biological Confocal Microscopy, J.Pawley, ed. (Plenum, 1995), pp. 491–505.
  8. 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]
  9. A. Esposito, H. C. Gerritsen, and F. S. Wouters, “Optimizing frequency-domain fluorescence lifetime sensing for high-throughput applications: photon economy and acquisition speed,” J. Opt. Soc. Am. A 24, 3261–3273 (2007).
    [CrossRef]
  10. B. R. Frieden, Probability, Statistical Optics, and Data Testing: A Problem Solving Approach (Springer, 2001).
    [CrossRef]
  11. J. Hayya and D. Armstrong, “A note on the ratio of two normally distributed variables,” Manage. Sci. 21, 1338–1341 (1975).
    [CrossRef]

2007 (2)

M. C. Skala, K. M. Riching, A. Gendron-Fitzpatrick, J. Eickhoff, K. W. Eliceiri, J. G. White, and N. Ramanujam, “In vivo multiphoton microscopy of NADH and FAD redox states, fluorescence lifetimes, and cellular morphology in precancerous epithelia,” Proc. Natl. Acad. Sci. U.S.A. 104, 19494–19499 (2007).
[CrossRef] [PubMed]

A. Esposito, H. C. Gerritsen, and F. S. Wouters, “Optimizing frequency-domain fluorescence lifetime sensing for high-throughput applications: photon economy and acquisition speed,” J. Opt. Soc. Am. A 24, 3261–3273 (2007).
[CrossRef]

2005 (1)

E. B. van Munster and T. W. J. Gadella, “Fluorescence lifetime imaging microscopy (FLIM),” Adv. Biochem. Eng./Biotechnol. 95, 143–175 (2005).

2003 (3)

T. Vo-Dinh, Biomedical Photonics Handbook (CRC, 2003).
[CrossRef]

H.-J. Lin, P. Herman, and J. R. Lakowicz, “Fluorescence lifetime-resolved pH imaging of living cells,” Cytometry, Part A 52A, 77–89 (2003).
[CrossRef]

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]

2002 (1)

K. M. Hanson, M. J. Behne, N. P. Barry, T. M. Mauro, E. Gratton, and R. M. Clegg, “Two-photon fluorescence lifetime imaging of the skin stratum corneum pH gradient,” Biophys. J. 83, 1682–1690 (2002).
[CrossRef] [PubMed]

2001 (1)

B. R. Frieden, Probability, Statistical Optics, and Data Testing: A Problem Solving Approach (Springer, 2001).
[CrossRef]

1995 (2)

A. Draaijer, R. Sanders, and H. C. Gerritsen, “Fluorescence lifetime imaging, a new tool in confocal microscopy” in Handbook of Biological Confocal Microscopy, J.Pawley, ed. (Plenum, 1995), pp. 491–505.

R. Sanders, A. Draaijer, H. C. Gerritsen, P. M. Houpt, and Y. K. Levine, “Quantitative pH imaging in cells using confocal fluorescence lifetime imaging microscopy,” Anal. Biochem. 227, 302–308 (1995).
[CrossRef] [PubMed]

1975 (1)

J. Hayya and D. Armstrong, “A note on the ratio of two normally distributed variables,” Manage. Sci. 21, 1338–1341 (1975).
[CrossRef]

Armstrong, D.

J. Hayya and D. Armstrong, “A note on the ratio of two normally distributed variables,” Manage. Sci. 21, 1338–1341 (1975).
[CrossRef]

Barry, N. P.

K. M. Hanson, M. J. Behne, N. P. Barry, T. M. Mauro, E. Gratton, and R. M. Clegg, “Two-photon fluorescence lifetime imaging of the skin stratum corneum pH gradient,” Biophys. J. 83, 1682–1690 (2002).
[CrossRef] [PubMed]

Behne, M. J.

K. M. Hanson, M. J. Behne, N. P. Barry, T. M. Mauro, E. Gratton, and R. M. Clegg, “Two-photon fluorescence lifetime imaging of the skin stratum corneum pH gradient,” Biophys. J. 83, 1682–1690 (2002).
[CrossRef] [PubMed]

Carlsson, K.

Clegg, R. M.

K. M. Hanson, M. J. Behne, N. P. Barry, T. M. Mauro, E. Gratton, and R. M. Clegg, “Two-photon fluorescence lifetime imaging of the skin stratum corneum pH gradient,” Biophys. J. 83, 1682–1690 (2002).
[CrossRef] [PubMed]

Draaijer, A.

R. Sanders, A. Draaijer, H. C. Gerritsen, P. M. Houpt, and Y. K. Levine, “Quantitative pH imaging in cells using confocal fluorescence lifetime imaging microscopy,” Anal. Biochem. 227, 302–308 (1995).
[CrossRef] [PubMed]

A. Draaijer, R. Sanders, and H. C. Gerritsen, “Fluorescence lifetime imaging, a new tool in confocal microscopy” in Handbook of Biological Confocal Microscopy, J.Pawley, ed. (Plenum, 1995), pp. 491–505.

Eickhoff, J.

M. C. Skala, K. M. Riching, A. Gendron-Fitzpatrick, J. Eickhoff, K. W. Eliceiri, J. G. White, and N. Ramanujam, “In vivo multiphoton microscopy of NADH and FAD redox states, fluorescence lifetimes, and cellular morphology in precancerous epithelia,” Proc. Natl. Acad. Sci. U.S.A. 104, 19494–19499 (2007).
[CrossRef] [PubMed]

Eliceiri, K. W.

M. C. Skala, K. M. Riching, A. Gendron-Fitzpatrick, J. Eickhoff, K. W. Eliceiri, J. G. White, and N. Ramanujam, “In vivo multiphoton microscopy of NADH and FAD redox states, fluorescence lifetimes, and cellular morphology in precancerous epithelia,” Proc. Natl. Acad. Sci. U.S.A. 104, 19494–19499 (2007).
[CrossRef] [PubMed]

Esposito, A.

Frieden, B. R.

B. R. Frieden, Probability, Statistical Optics, and Data Testing: A Problem Solving Approach (Springer, 2001).
[CrossRef]

Gadella, T. W. J.

E. B. van Munster and T. W. J. Gadella, “Fluorescence lifetime imaging microscopy (FLIM),” Adv. Biochem. Eng./Biotechnol. 95, 143–175 (2005).

Gendron-Fitzpatrick, A.

M. C. Skala, K. M. Riching, A. Gendron-Fitzpatrick, J. Eickhoff, K. W. Eliceiri, J. G. White, and N. Ramanujam, “In vivo multiphoton microscopy of NADH and FAD redox states, fluorescence lifetimes, and cellular morphology in precancerous epithelia,” Proc. Natl. Acad. Sci. U.S.A. 104, 19494–19499 (2007).
[CrossRef] [PubMed]

Gerritsen, H. C.

A. Esposito, H. C. Gerritsen, and F. S. Wouters, “Optimizing frequency-domain fluorescence lifetime sensing for high-throughput applications: photon economy and acquisition speed,” J. Opt. Soc. Am. A 24, 3261–3273 (2007).
[CrossRef]

A. Draaijer, R. Sanders, and H. C. Gerritsen, “Fluorescence lifetime imaging, a new tool in confocal microscopy” in Handbook of Biological Confocal Microscopy, J.Pawley, ed. (Plenum, 1995), pp. 491–505.

R. Sanders, A. Draaijer, H. C. Gerritsen, P. M. Houpt, and Y. K. Levine, “Quantitative pH imaging in cells using confocal fluorescence lifetime imaging microscopy,” Anal. Biochem. 227, 302–308 (1995).
[CrossRef] [PubMed]

Gratton, E.

K. M. Hanson, M. J. Behne, N. P. Barry, T. M. Mauro, E. Gratton, and R. M. Clegg, “Two-photon fluorescence lifetime imaging of the skin stratum corneum pH gradient,” Biophys. J. 83, 1682–1690 (2002).
[CrossRef] [PubMed]

Hanson, K. M.

K. M. Hanson, M. J. Behne, N. P. Barry, T. M. Mauro, E. Gratton, and R. M. Clegg, “Two-photon fluorescence lifetime imaging of the skin stratum corneum pH gradient,” Biophys. J. 83, 1682–1690 (2002).
[CrossRef] [PubMed]

Hayya, J.

J. Hayya and D. Armstrong, “A note on the ratio of two normally distributed variables,” Manage. Sci. 21, 1338–1341 (1975).
[CrossRef]

Herman, P.

H.-J. Lin, P. Herman, and J. R. Lakowicz, “Fluorescence lifetime-resolved pH imaging of living cells,” Cytometry, Part A 52A, 77–89 (2003).
[CrossRef]

Houpt, P. M.

R. Sanders, A. Draaijer, H. C. Gerritsen, P. M. Houpt, and Y. K. Levine, “Quantitative pH imaging in cells using confocal fluorescence lifetime imaging microscopy,” Anal. Biochem. 227, 302–308 (1995).
[CrossRef] [PubMed]

Lakowicz, J. R.

H.-J. Lin, P. Herman, and J. R. Lakowicz, “Fluorescence lifetime-resolved pH imaging of living cells,” Cytometry, Part A 52A, 77–89 (2003).
[CrossRef]

Levine, Y. K.

R. Sanders, A. Draaijer, H. C. Gerritsen, P. M. Houpt, and Y. K. Levine, “Quantitative pH imaging in cells using confocal fluorescence lifetime imaging microscopy,” Anal. Biochem. 227, 302–308 (1995).
[CrossRef] [PubMed]

Lin, H. -J.

H.-J. Lin, P. Herman, and J. R. Lakowicz, “Fluorescence lifetime-resolved pH imaging of living cells,” Cytometry, Part A 52A, 77–89 (2003).
[CrossRef]

Mauro, T. M.

K. M. Hanson, M. J. Behne, N. P. Barry, T. M. Mauro, E. Gratton, and R. M. Clegg, “Two-photon fluorescence lifetime imaging of the skin stratum corneum pH gradient,” Biophys. J. 83, 1682–1690 (2002).
[CrossRef] [PubMed]

Philip, J.

Ramanujam, N.

M. C. Skala, K. M. Riching, A. Gendron-Fitzpatrick, J. Eickhoff, K. W. Eliceiri, J. G. White, and N. Ramanujam, “In vivo multiphoton microscopy of NADH and FAD redox states, fluorescence lifetimes, and cellular morphology in precancerous epithelia,” Proc. Natl. Acad. Sci. U.S.A. 104, 19494–19499 (2007).
[CrossRef] [PubMed]

Riching, K. M.

M. C. Skala, K. M. Riching, A. Gendron-Fitzpatrick, J. Eickhoff, K. W. Eliceiri, J. G. White, and N. Ramanujam, “In vivo multiphoton microscopy of NADH and FAD redox states, fluorescence lifetimes, and cellular morphology in precancerous epithelia,” Proc. Natl. Acad. Sci. U.S.A. 104, 19494–19499 (2007).
[CrossRef] [PubMed]

Sanders, R.

R. Sanders, A. Draaijer, H. C. Gerritsen, P. M. Houpt, and Y. K. Levine, “Quantitative pH imaging in cells using confocal fluorescence lifetime imaging microscopy,” Anal. Biochem. 227, 302–308 (1995).
[CrossRef] [PubMed]

A. Draaijer, R. Sanders, and H. C. Gerritsen, “Fluorescence lifetime imaging, a new tool in confocal microscopy” in Handbook of Biological Confocal Microscopy, J.Pawley, ed. (Plenum, 1995), pp. 491–505.

Skala, M. C.

M. C. Skala, K. M. Riching, A. Gendron-Fitzpatrick, J. Eickhoff, K. W. Eliceiri, J. G. White, and N. Ramanujam, “In vivo multiphoton microscopy of NADH and FAD redox states, fluorescence lifetimes, and cellular morphology in precancerous epithelia,” Proc. Natl. Acad. Sci. U.S.A. 104, 19494–19499 (2007).
[CrossRef] [PubMed]

van Munster, E. B.

E. B. van Munster and T. W. J. Gadella, “Fluorescence lifetime imaging microscopy (FLIM),” Adv. Biochem. Eng./Biotechnol. 95, 143–175 (2005).

Vo-Dinh, T.

T. Vo-Dinh, Biomedical Photonics Handbook (CRC, 2003).
[CrossRef]

White, J. G.

M. C. Skala, K. M. Riching, A. Gendron-Fitzpatrick, J. Eickhoff, K. W. Eliceiri, J. G. White, and N. Ramanujam, “In vivo multiphoton microscopy of NADH and FAD redox states, fluorescence lifetimes, and cellular morphology in precancerous epithelia,” Proc. Natl. Acad. Sci. U.S.A. 104, 19494–19499 (2007).
[CrossRef] [PubMed]

Wouters, F. S.

Adv. Biochem. Eng./Biotechnol. (1)

E. B. van Munster and T. W. J. Gadella, “Fluorescence lifetime imaging microscopy (FLIM),” Adv. Biochem. Eng./Biotechnol. 95, 143–175 (2005).

Anal. Biochem. (1)

R. Sanders, A. Draaijer, H. C. Gerritsen, P. M. Houpt, and Y. K. Levine, “Quantitative pH imaging in cells using confocal fluorescence lifetime imaging microscopy,” Anal. Biochem. 227, 302–308 (1995).
[CrossRef] [PubMed]

Biophys. J. (1)

K. M. Hanson, M. J. Behne, N. P. Barry, T. M. Mauro, E. Gratton, and R. M. Clegg, “Two-photon fluorescence lifetime imaging of the skin stratum corneum pH gradient,” Biophys. J. 83, 1682–1690 (2002).
[CrossRef] [PubMed]

Cytometry, Part A (1)

H.-J. Lin, P. Herman, and J. R. Lakowicz, “Fluorescence lifetime-resolved pH imaging of living cells,” Cytometry, Part A 52A, 77–89 (2003).
[CrossRef]

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

Manage. Sci. (1)

J. Hayya and D. Armstrong, “A note on the ratio of two normally distributed variables,” Manage. Sci. 21, 1338–1341 (1975).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A. (1)

M. C. Skala, K. M. Riching, A. Gendron-Fitzpatrick, J. Eickhoff, K. W. Eliceiri, J. G. White, and N. Ramanujam, “In vivo multiphoton microscopy of NADH and FAD redox states, fluorescence lifetimes, and cellular morphology in precancerous epithelia,” Proc. Natl. Acad. Sci. U.S.A. 104, 19494–19499 (2007).
[CrossRef] [PubMed]

Other (3)

T. Vo-Dinh, Biomedical Photonics Handbook (CRC, 2003).
[CrossRef]

B. R. Frieden, Probability, Statistical Optics, and Data Testing: A Problem Solving Approach (Springer, 2001).
[CrossRef]

A. Draaijer, R. Sanders, and H. C. Gerritsen, “Fluorescence lifetime imaging, a new tool in confocal microscopy” in Handbook of Biological Confocal Microscopy, J.Pawley, ed. (Plenum, 1995), pp. 491–505.

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of homodyne lock-in detection for FD-FLIM.

Fig. 2
Fig. 2

Principle of FD-FLIM using sinusoidal excitation.

Fig. 3
Fig. 3

Comparison of Monte Carlo simulation, the rigorous PDF, and the approximate PDF for sinusoidal excitation. The cases for N = 10 , 50, and 1000 are shown in (a), (b), and (c), respectively. In each subfigure, the solid curve is the rigorous PDF of estimated lifetime, the dashed curve is the approximate PDF, and the filled circles represent the results of Monte Carlo simulation.

Fig. 4
Fig. 4

Comparison of Monte Carlo simulation, the rigorous PDF, and the approximate PDF for square-wave excitation. The cases for N = 10 , 50, and 1000 are shown in (a), (b), and (c), respectively. In each subfigure, the solid curve is the rigorous PDF of estimated lifetime, the dashed curve is the approximate PDF, and the filled circles represent the results of Monte Carlo simulation.

Fig. 5
Fig. 5

Comparison of Monte Carlo simulation, the rigorous PDF, and the approximate PDF for Dirac train excitation. The cases for N = 10 , 50, and 1000 are shown in (a), (b), and (c), respectively. In each subfigure, the solid curve is the rigorous PDF of estimated lifetime, the dashed curve is the approximate PDF, and the filled circles represent the results of Monte Carlo simulation.

Fig. 6
Fig. 6

F-value for square-wave excitation. (a) shows F-value as a function of duty cycle d and modulation frequency ω. The dashed line in (a) represents the position of the lowest F-value for each duty cycle. The F-value along the dashed line is plotted in (b).

Fig. 7
Fig. 7

F-value versus modulation frequency ω for Dirac train excitation.

Equations (78)

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I ( t ) = I 0   exp ( t / τ ) ,
F = σ τ ̂ τ N .
I em ( t ) = η c ( I ex I ) ( t ) ,
λ ( t ) = α η I em ( t ) .
λ i ( t ) = λ ( t ) sin ( ω t + ϕ i ) .
g i = ( λ i h ) ( t ) t = 0 ,
p g i ( g i ) = 1 2 π σ i 2 exp [ ( g i μ i ) 2 2 σ i 2 ] ,
p g 1 g 2 ( g 1 , g 2 ) = 1 2 π σ 1 σ 2 1 ρ 2 exp [ z 2 ( 1 ρ 2 ) ] ,
z = ( g 1 μ 1 ) 2 σ 1 2 2 ρ ( g 1 μ 1 ) ( g 2 μ 2 ) σ 1 σ 2 + ( g 2 μ 2 ) 2 σ 2 2 ,
P k ( k ) = a k k ! exp ( a ) ,
k = σ k 2 = a .
Φ k ( s ) = exp ( s k ) = k = 0 P k ( k ) exp ( s k ) = exp { a [ exp ( s ) 1 ] } .
k j = λ ¯ ( j Δ t ) Δ t α ,
λ ¯ ( t ) = λ ( t ) .
g 1 = j = k j α   sin ( ω j Δ t ) h ( t j Δ t ) t = 0 .
Φ g 1 ( s ) = exp [ s j = k j α   sin ( ω j Δ t ) h ( t j Δ t ) ] t = 0 = j = Φ k j [ s α   sin ( ω j Δ t ) h ( t j Δ t ) ] t = 0 ,
ln   Φ g 1 ( s ) = j = | λ ¯ ( j Δ t ) Δ t α { exp [ s α   sin ( ω j Δ t ) h ( t j Δ t ) ] 1 } | t = 0 .
ln   Φ g 1 ( s ) = | λ ¯ ( t ) α { exp [ s α   sin ( ω t ) h ( t t ) ] 1 } d t | t = 0 ,
Φ g 1 ( s ) = exp | ( λ ¯ ( t ) α { exp [ s α   sin ( ω t ) h ( t t ) ] 1 } d t ) | t = 0 .
μ 1 = g 1 = | s Φ g 1 ( s ) | s = 0 = [ λ ¯ ( t ) sin ( ω t ) ] h ( t ) t = 0 ,
σ 1 2 = g 1 2 ( g 1 ) 2 = | 2 s 2 Φ g 1 ( s ) | s = 0 μ 1 2 = α [ λ ¯ ( t ) sin 2 ( ω t ) ] h 2 ( t ) t = 0 ,
μ 2 = [ λ ¯ ( t ) cos ( ω t ) ] h ( t ) t = 0 ,
σ 2 2 = α [ λ ¯ ( t ) cos 2 ( ω t ) ] h 2 ( t ) t = 0 .
ρ = cov ( g 1 , g 2 ) σ 1 σ 2 ,
σ g 2 = σ 1 2 + σ 2 2 + 2   cov ( g 1 , g 2 ) ,
σ g 2 = | 2 α [ λ ¯ ( t ) sin 2 ( ω t + π 4 ) ] h 2 ( t ) | t = 0 ,
ρ = | α [ λ ¯ ( t ) sin ( 2 ω t ) ] h 2 ( t ) σ 1 σ 2 | t = 0 .
τ ̂ = f ( g 1 , g 2 ) .
F τ ̂ ( τ ̂ ) = P ( τ ̂ < τ ̂ ) ,
F τ ̂ ( τ ̂ ) = D τ ̂ p g 1 g 2 ( g 1 , g 2 ) d g 1 d g 2 ,
f ( g 1 , g 2 ) < τ ̂ .
I ex ( t ) = I ex 0 [ 1 + m ex   cos ( ω t ) ] ,
I em ( t ) = I em 0 [ 1 + m em   cos ( ω t ϕ ) ] ,
tan   ϕ = ω τ ,
m em = m ex 1 + ω 2 τ 2 .
λ ¯ ( t ) = α η I em 0 [ 1 + m em   cos ( ω t ϕ ) ] .
μ 1 = [ α η I em 0   sin ( ω t ) + 1 2 α η m em I em 0   cos ( 2 ω t ϕ ) + 1 2 α η m em I em 0   sin   ϕ ] h ( t ) t = 0 .
μ 1 = 1 2 α η m em I em 0   sin   ϕ h ( t ) d t = 1 2 α η m em I em 0   sin   ϕ H ( 0 ) ,
σ 1 2 = 1 2 α 2 η I em 0 [ 1 + m em   cos ( ω t ϕ ) + cos ( 2 ω t ) + m em   cos ( ω t ϕ ) cos ( 2 ω t ) ] h 2 ( t ) t = 0 .
σ 1 2 = 1 2 α 2 η I em 0 h 2 ( t ) d t .
h ( t ) = rect ( t T ) ,
rect ( x ) = { 1 , | x | 1 2 0 , otherwise . }
μ 1 = 1 2 α η m em I em 0   sin   ϕ T = 1 2 α m em N   sin   ϕ = α m ex N   sin   ϕ 2 1 + ω 2 τ 2 ,
σ 1 2 = 1 2 α 2 η I em 0 T = α 2 N 2 ,
H ( ν ) = 1 2 ν c rect ( ν 2 ν c ) ,
μ 1 = 1 2 α η m em I em 0   sin   ϕ 1 2 ν c ,
σ 1 2 = α 2 η I em 0 2 1 2 ν c .
μ 1 = 1 2 α m em N eff   sin   ϕ ,
σ 1 2 = α 2 N eff 2 ,
μ 2 = α m ex N   cos   ϕ 2 1 + ω 2 τ 2 ,
σ 2 2 = σ 1 2 = α 2 N 2 ,
ρ = 0.
τ ̂ = f ( g 1 , g 2 ) = 1 ω g 1 g 2 .
F τ ̂ ( τ ̂ ) = d g 2 ω τ ̂ g 2 p g 1 g 2 ( g 1 , g 2 ) d g 1 .
p τ ̂ ( τ ̂ ) = ω g 2 p g 1 g 2 ( ω τ ̂ g 2 , g 2 ) d g 2 = 1 + τ τ ̂ ω 2 ( 1 + ω 2 τ ̂ 2 ) 3 / 2 2 π σ 2 exp [ ( τ τ ̂ ) 2 1 + ω 2 τ ̂ 2 2 σ 2 ] ,
σ 2 = 2 ( 1 + τ 2 ω 2 ) 2 m ex 2 N ω 2 .
p τ ̂ ( τ ̂ ) = 1 2 π σ τ ̂ 2 exp [ ( τ ̂ τ ) 2 2 σ τ ̂ 2 ] ,
σ τ ̂ 2 = 2 ( 1 + τ 2 ω 2 ) 3 m ex 2 N ω 2 .
τ ̂ = τ ,
τ ̂ Taylor = 1 ω ( μ 1 μ 2 + σ 2 2 μ 1 μ 2 3 ρ σ 1 σ 2 μ 2 2 ) = τ [ 1 + 2 ( 1 + ω 2 τ 2 ) m ex 2 N ] ,
σ τ ̂ , Taylor 2 = 1 ω 2 ( σ 2 2 μ 1 2 μ 2 4 + σ 1 2 μ 2 2 2 ρ σ 1 σ 2 μ 1 μ 2 3 ) = 2 ( 1 + τ 2 ω 2 ) 3 m ex 2 N ω 2 .
F = 2 ( 1 + τ 2 ω 2 ) 3 m ex 2 τ 2 ω 2 .
I ex = I ex 0   rect ( t 2 π d / ω ) 1 2 π / ω comb ( t 2 π / ω ) ,
comb ( t ) = i = δ ( t i ) ,
μ 1 = N ω τ   sin ( π d ) π d ( 1 + ω 2 τ 2 ) ,
μ 2 = N   sin ( π d ) π d ( 1 + ω 2 τ 2 ) ,
σ 1 2 = N 4 [ 2 sin ( 2 π d ) π d ( 1 + 4 ω 2 τ 2 ) ] ,
σ 2 2 = N 4 [ 2 + sin ( 2 π d ) π d ( 1 + 4 ω 2 τ 2 ) ] ,
ρ = 2 ω τ   sin ( 2 π d ) 4 π 2 d 2 ( 1 + 4 ω 2 τ 2 ) 2 sin 2 ( 2 π d ) .
p τ ̂ ( τ ̂ ) = ω   sin ( π d ) [ 2 N ( 1 + 4 ω 2 τ 2 ) / d ] 1 / 2 π ( 1 + ω 2 τ 2 ) [ 4 π 2 d 2 ( 1 + 4 ω 2 τ 2 ) sin 2 ( 2 π d ) + 16 π 2 d 2 ω 2 τ 2 ( 1 + 4 ω 2 τ 2 ) 4 π 2 d 2 ( 1 + 4 ω 2 τ 2 ) 2 sin 2 ( 2 π d ) ] 1 / 2 { 2 π d ( 1 + 4 ω 2 τ 2 ) ( 1 + ω 2 τ ̂ τ ) [ 1 + ω 2 τ ( τ ̂ + 2 τ ) ] sin ( 2 π d ) } { 2 π d ( 1 + 4 ω 2 τ 2 ) ( 1 + ω 2 τ 2 ) + [ 1 + ω 2 τ ̂ ( τ ̂ 4 τ ) ] sin ( 2 π d ) } 3 / 2 exp ( 2 N ω 2 sin 2 ( π d ) ( 1 + 4 ω 2 τ 2 ) ( τ ̂ τ ) 2 π d ( 1 + ω 2 τ 2 ) 2 { 2 π d ( 1 + 5 ω 2 τ 2 + 4 ω 4 τ 4 ) + [ 1 + ω 2 τ ̂ ( τ ̂ 4 τ ) ] sin ( 2 π d ) } ) .
σ τ ̂ 2 = π d ( 1 + ω 2 τ 2 ) 2 [ 2 π d ( 1 + 5 ω 2 τ 2 + 4 ω 4 τ 4 ) ( 1 + 3 ω 2 τ 2 ) sin ( 2 π d ) ] 4 N ω 2 ( 1 + 4 ω 2 τ 2 ) sin 2 ( π d ) ,
F = { π d ( 1 + ω 2 τ 2 ) 2 [ 2 π d ( 1 + 5 ω 2 τ 2 + 4 ω 4 τ 4 ) ( 1 + 3 ω 2 τ 2 ) sin ( 2 π d ) ] 4 ω 2 τ 2 ( 1 + 4 ω 2 τ 2 ) sin 2 ( π d ) } 1 / 2 .
p τ ̂ ( τ ̂ ) = τ 2 ( 1 + 2 τ ̂ τ ω 2 ) N + 4 N ω 2 τ 2 2 π ( 1 + ω 2 τ 2 ) [ 2 τ 2 2 τ ̂ τ + τ ̂ 2 ( 1 + 2 ω 2 τ 2 ) ] 3 / 2 exp { N ( 1 + 4 ω 2 τ 2 ) ( τ ̂ τ ) 2 2 ( 1 + ω 2 τ 2 ) 2 [ 2 τ 2 2 τ ̂ τ + τ ̂ 2 ( 1 + 2 ω 2 τ 2 ) ] } ,
σ τ ̂ 2 = τ 2 ( 1 + 2 ω 2 τ 2 ) ( 1 + ω 2 τ 2 ) 2 N ( 1 + 4 ω 2 τ 2 ) .
F = [ ( 1 + 2 ω 2 τ 2 ) ( 1 + ω 2 τ 2 ) 2 ( 1 + 4 ω 2 τ 2 ) ] 1 / 2 ,
A = E F 2 k em n τ 1 ,
A CW = C 1 d F ( d , ω ) 2 ,
A Pulsed = C 2 ω F ( ω ) 2 ,

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