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  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 7, Iss. 6 — May. 25, 2012
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Short time behavior of fluorescence intensity fluctuations in single molecule polarization sensitive experiments

Lior Turgeman and Dror Fixler  »View Author Affiliations


Optics Express, Vol. 20, Issue 8, pp. 9276-9283 (2012)
http://dx.doi.org/10.1364/OE.20.009276


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Abstract

Recent developments in the field of single molecule orientation imaging have led us to devise a simple framework for analyzing fluorescence intensity fluctuations in single molecule polarization sensitive experiments. Based on the new framework, rotational dynamics of individual molecules are quantified, in this paper, from the short time behavior of the time averaged fluorescence intensity fluctuation trajectories. The suggested model can be applied in single molecule fluorescence fluctuations experiments to extract accurate expectation values of photon counts during very short integration time in which rotational diffusion is likely not to be averaged out.

© 2012 OSA

1. Introduction

Single molecule detection techniques have significantly progressed in recent years in the study of dynamic processes such as chemical and biological reaction kinetics and probing molecular motion in heterogeneous materials, without the loss of information that is typically encountered in ensemble averaging [1

1. C. Gell, D. Brockwell, and A. Smith, Handbook of single molecule fluorescence spectroscopy (Oxford University Press, 2006).

]. Detection of single molecule rotational motion is an exquisitely sensitive and useful tool in exploring the dynamics in various biological and chemical systems [2

2. A. P. Bartko, K. Xu, and R. M. Dickson, “Three-dimensional single molecule rotational diffusion in glassy state polymer films,” Phys. Rev. Lett. 89(2), 026101 (2002). [CrossRef] [PubMed]

4

4. X. Tan, D. Hu, T. C. Squier, and H. P. Lu, “Probing nanosecond protein motions of calmodulin by single-molecule fluorescence anisotropy,” Appl. Phys. Lett. 85(12), 2420–2422 (2004). [CrossRef]

]. A measure of dipole orientation can be obtained by calculating the reduced linear dichroism signal that is measured by detecting any two orthogonal polarizations in the in-plane of the sample and taking their difference divided by their sum. While in ensemble measurements the rotational dynamics is extracted by fitting the polarization signal to an exponential function [5

5. D. Fixler, R. Tirosh, A. Shainberg, and M. Deutsch, “Cytoplasmic changes in cardiac cells during contraction cycle detected by fluorescence polarization,” J. Fluoresc. 11(2), 89–100 (2001). [CrossRef]

], rotational dynamics of individual molecules is extracted by correlating the fluctuations of the reduced dichroism signal [6

6. G. Hinze, T. Basché, and R. A. L. Vallée, “Single molecule probing of dynamics in supercooled polymers,” Phys. Chem. Chem. Phys. 13(5), 1813–1818 (2011). [CrossRef] [PubMed]

]. However, most recent experiments have indicated deviations from the expected exponential behavior for the reduced linear dichroism correlation function in different observation times [7

7. L. A. Deschenes and D. A. Vanden Bout, “Heterogeneous dynamics and domains in supercooled o-Terphenyl: a single molecule study,” J. Phys. Chem. B 106(44), 11438–11445 (2002). [CrossRef]

, 8

8. R. Richert, “Heterogeneous dynamics in liquids: fluctuations in space and time,” J. Phys. Condens. Matter 14(23), R703–R738 (2002). [CrossRef]

]. Various approaches have been presented recently to analyze the origin of the non-exponential behavior for this signal [9

9. G. Hinze, G. Diezemann, and T. Basché, “Rotational correlation functions of single molecules,” Phys. Rev. Lett. 93(20), 203001 (2004). [CrossRef] [PubMed]

11

11. R. A. L. Vallée, T. Rohand, N. Boens, W. Dehaen, G. Hinze, and T. Basché, “Analysis of the exponential character of single molecule rotational correlation functions for large and small fluorescence collection angles,” J. Chem. Phys. 128(15), 154515 (2008). [CrossRef] [PubMed]

]. In any event, useful correlations require enhanced temporal resolution of experiment. In typical photon counting experiments, use of emission intensities for polarized light implies time averaging that increases the signal to noise ratio at the cost of temporal resolution. Thus, very fast fluorescence fluctuations may be obscured during integration time. Recently, several approaches have been presented aiming to increase temporal resolution of fluorescence signal [12

12. H. Yang and S. Xie, “Probing single-molecule dynamics photon by photon,” J. Chem. Phys. 117(24), 10965–10979 (2002). [CrossRef]

14

14. G. Hinze and T. Basché, “Statistical analysis of time resolved single molecule fluorescence data without time binning,” J. Chem. Phys. 132(4), 044509 (2010). [CrossRef] [PubMed]

].

This paper introduces a dynamic method to analyze fluorescence fluctuations in single molecule fluorescence polarization sensitive experiments. First principles of non-equilibrium analysis are used to derive a simple framework to extract rotational diffusion parameters from the short time behavior of time averaged fluorescence intensity fluctuations, while properly accounting for the impact of the numerical aperture (NA) value of the microscope objective and the timescale of fluorescence fluctuation.

2. Single molecule orientation imaging

This dependency is well-known and has been experimentally observed also in bulk measurements [16

16. D. Fixler, Y. Namer, Y. Yishay, and M. Deutsch, “Influence of fluorescence anisotropy on fluorescence intensity and lifetime measurement: theory, simulations and experiments,” IEEE Trans. Biomed. Eng. 53(6), 1141–1152 (2006). [CrossRef] [PubMed]

, 17

17. J. R. Lakowicz, Principles of fluorescence spectroscopy (Springer, 2006).

]. The single fluorophore is modeled by a dipole moment that is rapidly rotating in time. The effect of the polar angle ϕ on the measured fluorescence intensity (Imeasured) will drop as the molecule tips out of the sample plane.

3. Fluorescence intensity fluctuations equation

3.1 The influence of rotational diffusion on single molecule fluorescence intensity

Quantifying fluctuations of Imeasured as a result of ϕ rotation is of interest here. Combining Eq. (2) and Eq. (3) yields
ϕ(t)=sin-1Imeasured(t)2AIe2BIe=sin-1I˜(t)
(4)
where I˜(t) is a non-unit magnitude that takes into account Ie and the influence of the NA. We consider only changes in Imeasured, so the θ dependence in Eq. (5) (below) can be eliminated. Hence, the rotational diffusion equation for the averaged dipole distribution P reads [18

18. P. Debye, Polar molecules (Dover Publications, 1945).

]:
P(ϕ,t)t=Drot2P(ϕ,t)ϕ2
(5)
where P(ϕ,t) is the probability density function (PDF) of the dipole being oriented at a polar angle ϕ at time t, and Drot is the rotational diffusion constant. After inserting Eq. (4) into Eq. (5) and using rules of partial derivatives we find:
P(I˜,t)t=2Drot[(12I˜)P(I˜,t)I˜+2I˜(1I˜)P2(I˜,t)I˜2]
(6)
Equation (6) presented here for the first time – the core equation of this manuscript – provides a general framework for treating fluctuation of fluorescence intensity as a result of molecular reorientation. An equation for the averaged intensity is obtained by multiplying Eq. (6) withI˜(t), performing integration by parts, solving in Laplace space ts and inverting back to time domain
I˜(t)=12[1(2I˜0+1)e2t3τr]
(7)
Where Drot=16τr [18

18. P. Debye, Polar molecules (Dover Publications, 1945).

], τr is the rotational correlation time and I˜0=I˜(t=0). Consequently, Eq. (7) allows us to extract τr from an ensemble of fluorescence intensity trajectories that are measured for a single fluorophore. According to Eq. (7), higher values of τrare characterized by a slow convergence of I˜(t) as expected. Furthermore, in the long time limit I˜(t)1/2, which is a typical sin2ϕ behavior. However, convergence of I˜(t) is in the time scale of τr. Thus, in direct measurements of I˜(t), the temporal resolution used in experiments must be as low as τr itself in order to track fluctuations. Typically, using single photon counting devices, such as avalanche photodiodes or photomultiplier tubes used in photon counting experiments, leads to counting loss. Consequently, fast fluorescence fluctuations may be obscured during integration time.

3.2 Time averaged single molecule fluorescence intensity

3.3 Theory & Simulations

To gain further understanding of these analytical results and to test the effect of rotational diffusion, single molecule rotational trajectories at different correlation times were simulated using a random walk on a sphere, assuming θ(t=0)=π. Ten thousand angular trajectories were then used to compute fluorescence signals, and obtaining I˜¯(t) through Eq. (10). The simulation results (dotted lines) are compared to theory (solid lines) as can be seen in Fig. 2
Fig. 2 Simulations (dotted lines) versus theory (solid lines) for I¯(t) at different correlation times where and θ(t=0)=π, B≅C andIe=106ph/sec. The average is over 10,000 realizations of I¯(t). In the long time limit t→∞ we get the expected result I¯(t)=1/2.
. In the long time limit t→∞ we get the expected resultI˜¯(t)=1/2.

One can observe in Fig. 2 that for shorter values of τr, simulation deviates from theory (Eq. (10)) in the scale τr. This is a consequence of very fast rotational diffusion that yields less predictable value of I˜¯(t) for time bins of 1nsec used in simulations. Using a sampling time of 0.1 nsec, we get a perfect match (data not shown). That is, the size of the chosen time bins introduces an additional arbitrary time scale which should be short enough to track fluorescence intensity fluctuations. However, for each τr there is a time limit in which fluctuations become slower and can be tracked by larger time bins, as one can observe in (Fig. 2 – dashed rectangle).

4. The role of photon statistics in single molecule polarization sensitive experiments

In an actual experiment, the time limit in which fluctuations become slower can be quantitatively described by the photon counting distribution. For a stationary dipole, Poisson distribution fully describes the statistics of photon count detection. However, a change in dipole orientation will also cause a change in the mean Poisson quantity. If this change is random, then the arrival of photons to the detector, and hence their subsequent detection, is a doubly stochastic process.

The relation between continuous fluorescence intensity distribution and discrete photoelectron distribution was first derived by Mandel in the context of laser fluctuations [22

22. L. Mandel, “Fluctuations of photon beams and their correlations,” Proc. Phys. Soc. 72(6), 1037–1048 (1958). [CrossRef]

]:
P(n,T)=0Ωnn!eΩP(Ω)dΩ
(11)
where Ω is the sum of photons detected in time interval T. In the Mandel expression (11) one has to average the Poisson distributionΩnn!eΩ, according to the distribution of the time averaged fluorescence intensity. In the stationary stateP(Ω), and therefore alsoP(n,T), depend only on the length of the time interval T, but not on the time t itself. According to Eq. (11), fluctuations of the emitted fluorescence intensity will cause additional broadening of the photon counting distributionP(n,T) as a result of rotational diffusion. This broadening depends on the integration time T used in experiment. In the limit of long integration times (T), fluorescence intensity fluctuations will be completely averaged out in the corresponding fluctuations of Ω. In this case, the probability distributionP(Ω), approaches a delta function, and the P(n,T) will narrow to a Poissonian. For very short integration times (T0), fluctuations of Ωwill track the fluorescence intensity fluctuations I(t) completely. Thus, the probability distributions of Ωand I(t) are proportional to each other, P(Ω)=P(I)T and in order to capture intensity fluctuations of a particular process of interest characterized by P(I), one must choose an integration time T, shorter than the fluctuation time scale for that particular process. For this case the approximation, Ω˜~I˜shortTcan be used [23

23. D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, 1994).

25

25. Y. Jung, E. Barkai, and R. J. Silbey, “A stochastic theory of single molecule spectroscopy,” Adv. Chem. Phys. 123, 199–266 (2002). [CrossRef]

]:
P(n,T)0(IshortT)nn!eIshortTP(Ishort)dIshort
(12)
For simplicity, we take the detector area to be small enough so that the fluorescence intensity field I, is constant across the detector surface with a short sampling time interval of T. In experiment, very fast fluctuations resulting from rotational diffusion need to be tracked by choosing very short sampling intervals – as short as the effective T according to Eq. (12).

However, in the time limit in which Ω does not change significantly as result of slower fluctuations, larger sampling intervals can be used. In this extreme case which is τrdependent, P(n,T) is approximated by Poissonian with mean Ω(T) at time T:
P(n,T)Ω(T)nn!eΩ(T)
(13)
where Ω(T)=2BIeΩ˜(T)+ABT. This time limit, in which Eq. (13) is valid, can be determined by comparing the two approximations above (Eq. (12) and Eq. (13)). Whereas Eq. (12) determines the sampling time in which the fastest fluctuations are still tracked, Eq. (13) must yield the same effectiveP(n,T)for the chosen sampling time in order to continuously track slower fluctuations. For example, in case where n = 1.4, with high NA value of 1.3: ϕmax = 1.1905, A = 0.0781 and B = 0.04. One can observe in Fig. 3(a)
Fig. 3 Comparison of Eq. (12) for time interval T1 (solid line) and Eq. (13) for time interval T2 (dashed line), where τr = 2 ns. For NA = 1.3 and T1 = T2 = 10τrdetection probability for Eq. (12) is higher (a). For larger time interval of T2 = 100τr in Eq. (12), both approximations are identical (b). For NA = 0.4 detection probability is decreased for the latter case (c). Enlarging T2 to 1000τr increases detection probability for Eq. (12) (d).
that for τr=2ns, the realistic value for the rotational correlation time of fluorophore in water environment [17

17. J. R. Lakowicz, Principles of fluorescence spectroscopy (Springer, 2006).

] and T1 = T2 = 10τr (Eq. (12) for time interval T1 (solid line) and Eq. (13) for time interval T2 (dashed line)), Eq. (13) yields less effective P(n,T) than Eq. (12). Thus in this time limit of T2, fluctuations are not slow enough to be described by Eq. (13). While using T2 = 100τr in Eq. (13) and T1 = 10τrin Eq. (12), both approximations are almost identical (Fig. 3(b)). That is, by choosing different sampling rates in these different time scales of convergence, fluctuations are continuously tracked. However, in the latter case when lower NA value of 0.4 is used (leading to ϕmax = 0.2898, A = 0.000428 and B = 0.0098 values), P(n,T) is decreased as can be seen in Fig. 3(c). In this case, choosing T2 = 400τr increases detection probability (Fig. 3(d)). Obviously, for higher values of τr where convergence is slower, larger sampling intervals needs to be considered.

5. Discussion

In fluorescence fluctuation spectroscopy experiments, where it is essential to make sure that the chosen integration time in experiment is sufficiently short to track fluorescence intensity fluctuations, there are cases in which fluctuations, as a result of rotational diffusion, are not averaged out. Applying the suggested model in such cases should yield accurate expectation values of photon counts in very short integration time - as short as the rotational correlation time itself.

References and links

1.

C. Gell, D. Brockwell, and A. Smith, Handbook of single molecule fluorescence spectroscopy (Oxford University Press, 2006).

2.

A. P. Bartko, K. Xu, and R. M. Dickson, “Three-dimensional single molecule rotational diffusion in glassy state polymer films,” Phys. Rev. Lett. 89(2), 026101 (2002). [CrossRef] [PubMed]

3.

S. A. Rosenberg, M. E. Quinlan, J. N. Forkey, and Y. E. Goldman, “Rotational motions of macro-molecules by single-molecule fluorescence microscopy,” Acc. Chem. Res. 38(7), 583–593 (2005). [CrossRef] [PubMed]

4.

X. Tan, D. Hu, T. C. Squier, and H. P. Lu, “Probing nanosecond protein motions of calmodulin by single-molecule fluorescence anisotropy,” Appl. Phys. Lett. 85(12), 2420–2422 (2004). [CrossRef]

5.

D. Fixler, R. Tirosh, A. Shainberg, and M. Deutsch, “Cytoplasmic changes in cardiac cells during contraction cycle detected by fluorescence polarization,” J. Fluoresc. 11(2), 89–100 (2001). [CrossRef]

6.

G. Hinze, T. Basché, and R. A. L. Vallée, “Single molecule probing of dynamics in supercooled polymers,” Phys. Chem. Chem. Phys. 13(5), 1813–1818 (2011). [CrossRef] [PubMed]

7.

L. A. Deschenes and D. A. Vanden Bout, “Heterogeneous dynamics and domains in supercooled o-Terphenyl: a single molecule study,” J. Phys. Chem. B 106(44), 11438–11445 (2002). [CrossRef]

8.

R. Richert, “Heterogeneous dynamics in liquids: fluctuations in space and time,” J. Phys. Condens. Matter 14(23), R703–R738 (2002). [CrossRef]

9.

G. Hinze, G. Diezemann, and T. Basché, “Rotational correlation functions of single molecules,” Phys. Rev. Lett. 93(20), 203001 (2004). [CrossRef] [PubMed]

10.

C. Y. Wei, Y. H. Kim, R. K. Darst, P. J. Rossky, and D. A. Vanden Bout, “Origins of nonexponential decay in single molecule measurements of rotational dynamics,” Phys. Rev. Lett. 95(17), 173001 (2005). [CrossRef] [PubMed]

11.

R. A. L. Vallée, T. Rohand, N. Boens, W. Dehaen, G. Hinze, and T. Basché, “Analysis of the exponential character of single molecule rotational correlation functions for large and small fluorescence collection angles,” J. Chem. Phys. 128(15), 154515 (2008). [CrossRef] [PubMed]

12.

H. Yang and S. Xie, “Probing single-molecule dynamics photon by photon,” J. Chem. Phys. 117(24), 10965–10979 (2002). [CrossRef]

13.

R. A. L. Vallée, N. Tomczak, G. J. Vancso, L. Kuipers, and N. F. van Hulst, “Fluorescence lifetime fluctuations of single molecules probe local density fluctuations in disordered media: a bulk approach,” J. Chem. Phys. 122(11), 114704 (2005). [CrossRef] [PubMed]

14.

G. Hinze and T. Basché, “Statistical analysis of time resolved single molecule fluorescence data without time binning,” J. Chem. Phys. 132(4), 044509 (2010). [CrossRef] [PubMed]

15.

J. T. Fourkas, “Rapid determination of the three-dimensional orientation of single molecules,” Opt. Lett. 26(4), 211–213 (2001). [CrossRef] [PubMed]

16.

D. Fixler, Y. Namer, Y. Yishay, and M. Deutsch, “Influence of fluorescence anisotropy on fluorescence intensity and lifetime measurement: theory, simulations and experiments,” IEEE Trans. Biomed. Eng. 53(6), 1141–1152 (2006). [CrossRef] [PubMed]

17.

J. R. Lakowicz, Principles of fluorescence spectroscopy (Springer, 2006).

18.

P. Debye, Polar molecules (Dover Publications, 1945).

19.

H. Risken, The fokker-planck equation: methods of solutions and applications (Springer, 1980).

20.

M. Zwanziger and M. Lax, “Exact photocount distributions for lasers near threshold,” Phys. Rev. Lett. 24(17), 937–940 (1970). [CrossRef]

21.

L. Turgeman, S. Carmi, and E. Barkai, “Fractional Feynman-Kac equation for non-brownian functionals,” Phys. Rev. Lett. 103(19), 190201 (2009). [CrossRef] [PubMed]

22.

L. Mandel, “Fluctuations of photon beams and their correlations,” Proc. Phys. Soc. 72(6), 1037–1048 (1958). [CrossRef]

23.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, 1994).

24.

Y. Chen, J. D. Müller, P. T. C. So, and E. Gratton, “The photon counting histogram in fluorescence fluctuation spectroscopy,” Biophys. J. 77(1), 553–567 (1999). [CrossRef] [PubMed]

25.

Y. Jung, E. Barkai, and R. J. Silbey, “A stochastic theory of single molecule spectroscopy,” Adv. Chem. Phys. 123, 199–266 (2002). [CrossRef]

OCIS Codes
(000.5490) General : Probability theory, stochastic processes, and statistics
(030.5260) Coherence and statistical optics : Photon counting
(110.4280) Imaging systems : Noise in imaging systems
(180.2520) Microscopy : Fluorescence microscopy
(260.5430) Physical optics : Polarization
(270.5290) Quantum optics : Photon statistics

ToC Category:
Microscopy

History
Original Manuscript: February 2, 2012
Revised Manuscript: April 2, 2012
Manuscript Accepted: April 2, 2012
Published: April 6, 2012

Virtual Issues
Vol. 7, Iss. 6 Virtual Journal for Biomedical Optics

Citation
Lior Turgeman and Dror Fixler, "Short time behavior of fluorescence intensity fluctuations in single molecule polarization sensitive experiments," Opt. Express 20, 9276-9283 (2012)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-20-8-9276


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References

  1. C. Gell, D. Brockwell, and A. Smith, Handbook of single molecule fluorescence spectroscopy (Oxford University Press, 2006).
  2. A. P. Bartko, K. Xu, and R. M. Dickson, “Three-dimensional single molecule rotational diffusion in glassy state polymer films,” Phys. Rev. Lett.89(2), 026101 (2002). [CrossRef] [PubMed]
  3. S. A. Rosenberg, M. E. Quinlan, J. N. Forkey, and Y. E. Goldman, “Rotational motions of macro-molecules by single-molecule fluorescence microscopy,” Acc. Chem. Res.38(7), 583–593 (2005). [CrossRef] [PubMed]
  4. X. Tan, D. Hu, T. C. Squier, and H. P. Lu, “Probing nanosecond protein motions of calmodulin by single-molecule fluorescence anisotropy,” Appl. Phys. Lett.85(12), 2420–2422 (2004). [CrossRef]
  5. D. Fixler, R. Tirosh, A. Shainberg, and M. Deutsch, “Cytoplasmic changes in cardiac cells during contraction cycle detected by fluorescence polarization,” J. Fluoresc.11(2), 89–100 (2001). [CrossRef]
  6. G. Hinze, T. Basché, and R. A. L. Vallée, “Single molecule probing of dynamics in supercooled polymers,” Phys. Chem. Chem. Phys.13(5), 1813–1818 (2011). [CrossRef] [PubMed]
  7. L. A. Deschenes and D. A. Vanden Bout, “Heterogeneous dynamics and domains in supercooled o-Terphenyl: a single molecule study,” J. Phys. Chem. B106(44), 11438–11445 (2002). [CrossRef]
  8. R. Richert, “Heterogeneous dynamics in liquids: fluctuations in space and time,” J. Phys. Condens. Matter14(23), R703–R738 (2002). [CrossRef]
  9. G. Hinze, G. Diezemann, and T. Basché, “Rotational correlation functions of single molecules,” Phys. Rev. Lett.93(20), 203001 (2004). [CrossRef] [PubMed]
  10. C. Y. Wei, Y. H. Kim, R. K. Darst, P. J. Rossky, and D. A. Vanden Bout, “Origins of nonexponential decay in single molecule measurements of rotational dynamics,” Phys. Rev. Lett.95(17), 173001 (2005). [CrossRef] [PubMed]
  11. R. A. L. Vallée, T. Rohand, N. Boens, W. Dehaen, G. Hinze, and T. Basché, “Analysis of the exponential character of single molecule rotational correlation functions for large and small fluorescence collection angles,” J. Chem. Phys.128(15), 154515 (2008). [CrossRef] [PubMed]
  12. H. Yang and S. Xie, “Probing single-molecule dynamics photon by photon,” J. Chem. Phys.117(24), 10965–10979 (2002). [CrossRef]
  13. R. A. L. Vallée, N. Tomczak, G. J. Vancso, L. Kuipers, and N. F. van Hulst, “Fluorescence lifetime fluctuations of single molecules probe local density fluctuations in disordered media: a bulk approach,” J. Chem. Phys.122(11), 114704 (2005). [CrossRef] [PubMed]
  14. G. Hinze and T. Basché, “Statistical analysis of time resolved single molecule fluorescence data without time binning,” J. Chem. Phys.132(4), 044509 (2010). [CrossRef] [PubMed]
  15. J. T. Fourkas, “Rapid determination of the three-dimensional orientation of single molecules,” Opt. Lett.26(4), 211–213 (2001). [CrossRef] [PubMed]
  16. D. Fixler, Y. Namer, Y. Yishay, and M. Deutsch, “Influence of fluorescence anisotropy on fluorescence intensity and lifetime measurement: theory, simulations and experiments,” IEEE Trans. Biomed. Eng.53(6), 1141–1152 (2006). [CrossRef] [PubMed]
  17. J. R. Lakowicz, Principles of fluorescence spectroscopy (Springer, 2006).
  18. P. Debye, Polar molecules (Dover Publications, 1945).
  19. H. Risken, The fokker-planck equation: methods of solutions and applications (Springer, 1980).
  20. M. Zwanziger and M. Lax, “Exact photocount distributions for lasers near threshold,” Phys. Rev. Lett.24(17), 937–940 (1970). [CrossRef]
  21. L. Turgeman, S. Carmi, and E. Barkai, “Fractional Feynman-Kac equation for non-brownian functionals,” Phys. Rev. Lett.103(19), 190201 (2009). [CrossRef] [PubMed]
  22. L. Mandel, “Fluctuations of photon beams and their correlations,” Proc. Phys. Soc.72(6), 1037–1048 (1958). [CrossRef]
  23. D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, 1994).
  24. Y. Chen, J. D. Müller, P. T. C. So, and E. Gratton, “The photon counting histogram in fluorescence fluctuation spectroscopy,” Biophys. J.77(1), 553–567 (1999). [CrossRef] [PubMed]
  25. Y. Jung, E. Barkai, and R. J. Silbey, “A stochastic theory of single molecule spectroscopy,” Adv. Chem. Phys.123, 199–266 (2002). [CrossRef]

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