## The optics and performance of dual-focus fluorescence correlation spectroscopy

Optics Express, Vol. 16, Issue 19, pp. 14353-14368 (2008)

http://dx.doi.org/10.1364/OE.16.014353

Acrobat PDF (774 KB)

### Abstract

Fluorescence correlation spectroscopy (FCS) is an important spectroscopic technique which can be used for measuring the diffusion and thus size of fluorescing molecules at pico- to nanomolar concentrations. Recently, we introduced an extension of conventional FCS, which is called dual-focus FCS (2fFCS) and allows absolute diffusion measurements with high precision and repeatability. It was shown experimentally that the method is robust against most optical and sample artefacts which are troubling conventional FCS measurements, and is furthermore able to yield absolute values of diffusion coefficients without referencing against known standards. However, a thorough theoretical treatment of the performance of 2fFCS is still missing. The present paper aims at filling this gap. Here, we have systematically studied the performance of 2fFCS with respect to the most important optical and photophysical factors such as cover slide thickness, refractive index of the sample, laser beam geometry, and optical saturation. We show that 2fFCS has indeed a superior performance when compared with conventional FCS, being mostly insensitive to most potential aberrations when working under optimized conditions.

© 2008 Optical Society of America

## 1. Introduction

1. D. Magde, E. Elson, and W. W. Webb “Thermodynamic fluctuations in a reacting system - measurement by fluorescence correlation spectroscopy,“ Phys. Rev. Lett. **29**, 705–708, (1972). [CrossRef]

3. D. Magde, E. Elson, and W. W. Webb “Fluorescence Corelation Spectroscopy II. An Experimental Realization,” Biopolymers **13**, 29–61 (1974). [CrossRef] [PubMed]

4. J. Widengren and Ü. Mets*Single-Molecule Detection in Solution - Methods and Applications*, Eds.
C. Zander, J. Enderlein, and R. A. Keller
(Wiley-VCH, 2002) pp. 69–95. [CrossRef]

5. R. Rigler and E. Elson, Eds. *Fluorescence Correlation Spectroscopy* (Springer, 2001). [CrossRef]

6. A. Benda, M. Benes, V. Marecek, A. Lhotsky, W.T. Hermens, and M. Hof, “How To Determine Diffusion Coefficients in Planar Phospholipid Systems by Confocal Fluorescence Correlation Spectroscopy,” Langmuir **19**, 4120–4126 (2003). [CrossRef]

8. J. Ries and P. Schwille, “Studying Slow Membrane Dynamics with Continuous Wave Scanning Fluorescence Correlation Spectroscopy,” Biophys. J. **91**, 1915–1924 (2006). [CrossRef] [PubMed]

9. Z. Petrasek and P. Schwille, “Precise measurement of diffusion coefficients using scanning fluorescence correlation spectroscopy,” Biophys. J. **94**, 1437–1448 (2008). [CrossRef]

10. T. Dertinger, V. Pacheco, I. von der Hocht, R. Hartmann, I. Gregor, and J. Enderlein, “Two-focus fluorescence correlation spectroscopy: A new tool for accurate and absolute diffusion measurements,” ChemPhysChem **8**, 433–443 (2007). [CrossRef] [PubMed]

10. T. Dertinger, V. Pacheco, I. von der Hocht, R. Hartmann, I. Gregor, and J. Enderlein, “Two-focus fluorescence correlation spectroscopy: A new tool for accurate and absolute diffusion measurements,” ChemPhysChem **8**, 433–443 (2007). [CrossRef] [PubMed]

11. A. Loman, T. Dertinger, F. Koberling, and J. Enderlein, “Comparison of optical saturation effects in conventional and dual-focus fluorescence correlation spectroscopy,” Chem. Phys. Lett. **459**, 18–21 (2008). [CrossRef]

## 2. Theoretical background

### 2.1 Calculation of the molecule detection function

12. M. Böhmer, F. Pampaloni, M. Wahl, H. J. Rahn, R. Erdmann, and J. Enderlein, “Time-resolved confocal scanning device for ultrasensitive fluorescence detection,” Rev. Sci. Instrum . **72**, 4145–4152 (2001). [CrossRef]

13. B. K. Müller, E. Zaychikov, C. Bräuchle, and D. C. Lamb, “Pulsed interleaved excitation,” Biophys. J. **89**, 3508–3522 (2005). [CrossRef] [PubMed]

24. J. Enderlein, I. Gregor, D. Patra, T. Dertinger, and U. B. Kaupp, “Performance of Fluorescence Correlation Spectroscopy for Measuring Diffusion and Concentration,” ChemPhysChem **6**, 2324–2336 (2005). [CrossRef] [PubMed]

*I*

*(r) as a function position*

_{ex}**r**is calculated in a standard way following the seminal works by Richards and Wolf [15

15. E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. Roy. Soc. London A **253**, 349–357 (1959). [CrossRef]

16. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. London A **253**, 358–379 (1959). [CrossRef]

17. P. R. T. Munro and P. Török, “Vectorial, high numerical aperture study of Nomarski's differential interference contrast microscope,” Opt. Express **13**, 6833–47 (2005). [CrossRef] [PubMed]

18. P. Török, Z. Varga, G. R. Laczik, and J. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A **12**, 325 (1995). [CrossRef]

25. M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express **14**, 11277–11291 (2006). [CrossRef] [PubMed]

*I*

*(*

_{ex}**r**) and the dipole vector

**p**of the light-absorbing molecule. However, for calculating the ACF, one needs the average excitation probability of a molecule at a given position, and this probability depends also on optical saturation. Optical saturation occurs when the excitation intensity becomes so large that the molecule spends more and more time in a non-excitable state, so that increasing the excitation intensity does not lead to a proportional increase in emitted fluorescence intensity, see. e.g. [26

26. I. Gregor, D. Patra, and J. Enderlein, “Optical Saturation in Fluorescence Correlation Spectroscopy under Continuous-Wave and Pulsed Excitation,” ChemPhysChem **6**, 164–70 (2005). [CrossRef] [PubMed]

*cis-trans-*isomerization in cyanine dyes, or the optically induced dark states in quantum dots. In Ref.[26

26. I. Gregor, D. Patra, and J. Enderlein, “Optical Saturation in Fluorescence Correlation Spectroscopy under Continuous-Wave and Pulsed Excitation,” ChemPhysChem **6**, 164–70 (2005). [CrossRef] [PubMed]

*I*

*(*

_{em}**r**) of a molecule as a function of its position

**r**when the excitation intensity distribution

*I*

*(*

_{ex}**r**) is known. Again, we assumed that molecular rotational diffusion is much faster than the time scale of average excitation and emission rate when calculating the relation between

*I*

*(*

_{em}**r**) and

*I*

*(*

_{ex}**r**) following Ref.[26

26. I. Gregor, D. Patra, and J. Enderlein, “Optical Saturation in Fluorescence Correlation Spectroscopy under Continuous-Wave and Pulsed Excitation,” ChemPhysChem **6**, 164–70 (2005). [CrossRef] [PubMed]

**r**. This can be calculated by integrating the Poynting energy flux over the aperture of the confocal pinhole as induced by the emission of a molecule positioned at position

**r**. Taking again into account fast rotational diffusion, the CEF is obtained by averaging the Poynting energy flux through the confocal aperture over all possible orientations of the emitting molecule. The technical details of CEF calculations have been published several times, and the reader is referred to e.g. Ref.[24

24. J. Enderlein, I. Gregor, D. Patra, T. Dertinger, and U. B. Kaupp, “Performance of Fluorescence Correlation Spectroscopy for Measuring Diffusion and Concentration,” ChemPhysChem **6**, 2324–2336 (2005). [CrossRef] [PubMed]

27. P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. **45**, 1681–1698 (1998). [CrossRef]

30. M. Leutenegger and T. Lasser, “Detection efficiency in total internal reflection fluorescence microscopy,” Opt. Express **16**, 8519–8531 (2008). [CrossRef] [PubMed]

*U*(

**r**) is given by the product of the emission rate

*I*

*(*

_{em}**r**) (as calculated from the excitation rate

*I*

*(*

_{ex}**r**) taking into account optical saturation) times the CEF. The MDF is directly dependent on the position of the molecule in sample space, and indirectly on the excitation and emission conditions. For our numerical calculations, it is convenient to represent the MDF as a Fourier series over the angular variable φ (angle around the optical axis of the microscope) as

*z*) represent cylindrical coordinates with the

*z*-axis along the optical axis [24

24. J. Enderlein, I. Gregor, D. Patra, T. Dertinger, and U. B. Kaupp, “Performance of Fluorescence Correlation Spectroscopy for Measuring Diffusion and Concentration,” ChemPhysChem **6**, 2324–2336 (2005). [CrossRef] [PubMed]

*m*| < 10 in Eq. (1). Any inclusion of higher Fourier components did not affect the final results.

## 2.2 Calculation of autocorrelation and cross-correlation function

*g*(τ), is equivalent to determining the probability of detecting a photon at time

*t*+τ if there had been a photon detection event at time

*t*. As has been shown in detail in Ref.[24

**6**, 2324–2336 (2005). [CrossRef] [PubMed]

*F*

*is given by*

_{m}*c*is their concentration, δ

*,*

_{m}*is Kronecker’s symbol being unity for*

_{n}*m*=

*n*and zero otherwise, and

*J*

*denotes Bessel functions of the first kind. The integrations in the above equations have to be done numerically. Because the MDF falls off rapidly to zero when moving away from the focus centre, the integrations converge rather quickly to a final value when numerically integrating over larger and larger values of ρ and*

_{m}*z*.

*t*+τ from the second focus if there had been a photon detection event at time

*t*from the first focus, is done totally analogously to Eqs. (2) and (3), but by calculating first the

*F*

*using the MDF of the first focus, and evaluating then the integral in Eq. (2) by using the MDF of the second focus.*

_{m}## 2.3 Fitting of the correlation functions and extraction of diffusion coefficients

10. T. Dertinger, V. Pacheco, I. von der Hocht, R. Hartmann, I. Gregor, and J. Enderlein, “Two-focus fluorescence correlation spectroscopy: A new tool for accurate and absolute diffusion measurements,” ChemPhysChem **8**, 433–443 (2007). [CrossRef] [PubMed]

*x*and

*y*are transversal coordinates perpendicular to the optical axis

*z*=0, δ is the lateral distance between both foci, so that one focus is shifted by +δ/2 and the other by -δ/2 away from the optical axis along the

*x*-axis, and the functions κ(

*z*) and w(

*z*) are given by

*R*(

*z*) is defined by:

*is the excitation wavelength, λ*

_{ex}*the center emission wavelength,*

_{em}*n*is the refractive index of the immersion medium (water),

*a*is the radius of the confocal aperture divided by magnification, and

*w*

_{0}and

*R*

_{0}are two (generally unknown) model parameters.

*g*

*(τ)=*

_{ACF}*g*(τ,0), and the CCF is given by

*g*

*(τ)=*

_{CCF}*g*(τ,δ). We will use Eq. (8) for globally fitting the ACFs and CCF as calculated by Eqs. (2) and (3) using the exact MDF. For this fit, the exact ACFs and CCF are calculated on a logarithmic time scale (i.e. at logarithmically spaced τ-values, similar to real experiments where correlation functions are calculated by a multiple-tau algorithm, see e.g. Ref.[31

31. M. Wahl, I. Gregor, M. Patting, and J. Enderlein, “Fast calculation of fluorescence correlation data with asynchronous time-correlated single-photon counting,” Opt. Express **11**, 3583–91 (2003). [CrossRef] [PubMed]

*D*, the waist parameter

*w*

_{0}, and the confocal pinhole parameter

*R*

_{0}. The shear distance δ of the DIC prism is assumed to be known

*a priori*and is set equal to 400 nm in the following calculations.

## 3. Results and discussion

### 3.1 Anatomy of the Molecule Detection Function

*U*

*(ρ,*

_{m}*z*) is done on a square (ρ,

*z*)-grid with a grid spacing of λ

*/30. The grid extension is chosen large enough so that the MDFs for both foci have fallen everywhere below 10*

_{ex}^{-3}of their maximum values. Further refinement or extension of the grid size did not change the final results. All integrations in Eqs. (2) and (3) were carried out using finite element summation.

*e*

^{2}-beam-radius values of 1.25, 1.5, 2. 3 and 4 mm, thus starting with a rather large focus size and ending with a focus close to the diffraction limit (with the objective’s parameters as specified above, the radius of its back focal aperture is equal to 3.42 mm). The resulting overlapping MDFs are visualized in Fig. 2.

*w*(

*z*) of the Gaussian distributions as a function of the z-coordinate is shown in the top right panels of Figs. 3(a) and (b), together with a fit of Eq. (5). Furthermore, we considered the

*z*-dependence of the MDF, i.e.

*U*(ρ=0,

*z*), multiplied this function by

*w*

^{2}(

*z*), and fitted the resulting function with the expression from Eq. (6) (bottom right panels of Figs. 3(a) and (b). Additionally, we calculated with the MDFs the corresponding ACFs and CCF and fitted them with a global fit using Eq. (8), thus extracting values for the parameters

*w*

_{0}and

*R*

_{0}. We plotted in the right panels of Figs. 3(a) and (b) also the functions of Eq. (5) and Eq. (6) using these parameters as extracted from the 2fFCS fits. In the case of relaxed focusing, Fig. 4(a), the wave-optically calculated functions

*w*(

*z*) and κ(

*z*) can be perfectly fitted by the model curves of Eq. (5) and Eq. (6), and moreover, these fits are in perfect agreement with the parameters

*w*

_{0}and

*R*

_{0}as extracted from a 2fFCS measurement fit.

*w*

_{0}and

*R*

_{0}parameters as derived from fitting a 2fFCS measurement.

*z*), the resulting curve is clearly different from the actual situation. Thus, although a 2fFCS fit yields remarkably good estimates for the structure of the MDF at relaxed focusing conditions, its description of κ(

*z*) gets increasingly worse when approaching diffraction-limited focusing. Thus, it is now important to ask how well a 2fFCS will estimate an absolute value of a diffusion coefficient under different focusing conditions, in particular when knowing that with coming closer to diffraction-limited focusing the extracted fit value of

*R*

_{0}does not well describe the real

*z*-dependency of κ(

*z*). To achieve this, we simulated 2fFCS for different focusing conditions (i.e. by changing the laser beam diameter) and fitted the resulting ACFs and CCF with Eq. (8) for extracting absolute values of the diffusion coefficient

*D*. In the modelling we assumed that the sample molecules have some arbitrary diffusion coefficient of 5·10

^{-5}cm

^{2}/s. However, this absolute values is rather unimportant because we will consider always the ratio between fitted and actual value of a diffusion coefficient, which will be independent on absolute values.

*absolute*value of the diffusion coefficient from a 2fFCS measurement, assuming that one knows the distance between foci as introduced by the DIC prism exactly (which can be measured with high precision, see Ref.[32

32. C. B. Müller, K. Weiß, W. Richtering, A. Loman, and J. Enderlein, “Calibrating Differential Interference Contrast Microscopy with dual-focus Fluorescence Correlation Spectroscopy,” Opt. Express **16**, 4322–9 (2008). [CrossRef] [PubMed]

## 3.2 Laser astigmatism and ellipticity

*any*effect on the accuracy of determining diffusion coefficients from 2fFCS measurements, in stark contrast to what happens for conventional single-focus FCS [24

**6**, 2324–2336 (2005). [CrossRef] [PubMed]

## 3.3 Cover slide thickness deviation and refractive index mismatch

*not*yield

*absolute*values of diffusion coefficients but must always be referenced against a sample of known diffusion – in Fig. 5, we have taken the FCS value at zero thickness deviation as the reference for all the shown single-focus FCS values. As shown, conventional FCS is much more sensitive to this kind of aberration than 2fFCS. Remarkably, there is nearly no dependence of the determined 2fFCS value on cover-slide thickness (or, similarly, refractive index mismatch) when using a laser beam radius below ~2 mm. This is in perfect accordance with experimental results as reported in Ref.[10

**8**, 433–443 (2007). [CrossRef] [PubMed]

11. A. Loman, T. Dertinger, F. Koberling, and J. Enderlein, “Comparison of optical saturation effects in conventional and dual-focus fluorescence correlation spectroscopy,” Chem. Phys. Lett. **459**, 18–21 (2008). [CrossRef]

## 3.4 Optical Saturation

_{0}→ S

_{1}transition and the finite lifetime of the excited state. We explore the saturation behaviour for a measurement with pulsed excitation using a pulse width of 0.025

*τ*

*and repetition period of 12.5*

_{f}*τ*

*in units of the excited state lifetime*

_{f}*τ*

*.*

_{f}*I*

*=(σ·*

_{sat}*τ*

*)*

_{f}^{-1}(given here in units of photons per area per time), where σ denotes the molecules’ absorption cross section at the excitation wavelength, see Ref.[26

**6**, 164–70 (2005). [CrossRef] [PubMed]

*I*

*. The impact of varying saturation level on the apparent diffusion coefficient Dfit as fitted from a corresponding 2fFCS measurement is shown in Fig. 7. There, we compare the sensitivity of 2fFCS against S*

_{sat}_{0}→ S

_{1}optical saturation for different degrees of focusing. As can be seen, for rather relaxed focusing (laser beam radius below ~2 mm), the method is rather insensitive to saturation as long as the maximum excitation intensity remains below ~0.2

*I*

*. But even for rather extreme saturation values, the relative error is not larger than 7 %, again in stark contrast to conventional single-focus FCS as was analyzed in Ref.[24*

_{sat}**6**, 2324–2336 (2005). [CrossRef] [PubMed]

*I*

*. The considered laser beam radius was 2 mm, i.e. rather relaxed focusing.*

_{sat}_{0}→S

_{1}saturation is that, although the MDF can be heavily deformed by it, it does

*not*change the distance between the foci centres. Thus again, a global fit of ACF and CCF can mostly compensate for the effects introduced by saturation. However, besides the omnipresent S

_{0}→S

_{1}saturation of fluorescence, many molecules also exhibit more complex mechanisms of saturation, for example by being pumped into a nonfluorescent triplet state or into some other non-fluorescent conformation. In this instance, it may be expected that saturation has a much stronger impact also on 2fFCS.

*k*

*and triplet-state-relaxation rate constant*

_{isc}*k*

*, i.e. κ=k*

_{ph}*/k*

_{isc}*. Figure 8 shows a comparison of the performance of conventional FCS and 2fFCS as a function of excitation intensity in units of*

_{ph}*I*

*for different values of κ and an assumed laser beam radius of 2 mm. As can be seen now, with increasing triplet state pumping efficiency, the outcome of a 2fFCS measurement for the diffusion coefficient becomes more and more biased towards smaller values, although the sensitivity is still not as large as in the case of conventional FCS. Interestingly, we did not find this effect in measurements on Cy5, where one has optical saturation due to a very similar optically driven process of*

_{sat}*cis-trans*isomerization, see e.g. Ref.[11

11. A. Loman, T. Dertinger, F. Koberling, and J. Enderlein, “Comparison of optical saturation effects in conventional and dual-focus fluorescence correlation spectroscopy,” Chem. Phys. Lett. **459**, 18–21 (2008). [CrossRef]

33. C. B. Müller, A. Loman, V. Pacheco, F. Koberling, D. Willbold, W. Richtering, and J. Enderlein, “Precise Measurement of Diffusion by Multi-Color Dual-Focus Fluorescence Correlation Spectroscopy,” Eur. Phys. Lett. **83**, 46001 (2008). [CrossRef]

## 3.5 Detection volume

*V*

*except by calibrating it, accounting for all of the previously mentioned optical and photophysical problems that also trouble FCS as a method for precise diffusion measurements. 2fFCS also has the ability to deliver, in addition to the diffusion coefficient*

_{eff}*D*, values for the MDF parameters

*w*

_{0}and

*R*

_{0}, which then could be used to directly calculate

*V*

*, using Eq. (4) in Eq. (9). As an example demonstrating how accurate this would be, we compared the real value of*

_{eff}*V*

*(calculated directly from the exactly known MDF) with that which one obtains by using the model MDF Eq. (4) and the parameters*

_{eff}*w*

_{0}and

*R*

_{0}as extracted from fitting the corresponding 2fFCS curves. The result, calculated for different focusing conditions and different cover-slide thickness deviations (as a typical source of aberration), is shown in Fig. 9. As can be seen, the performance of 2fFCS in determining absolute concentrations of molecule is much worse than its ability to yield correct values of diffusion coefficients. The reason is that the temporal decay of the ACF/CCF is obviously much less sensitive to the details of the outbound regions of the MDF, whereas the detection volume is still quite sensitive to the details of the MDF even at positions rather far away from the focal plane or the optical axis.

## 4. Conclusion

## Acknowledgment

## References and links

1. | D. Magde, E. Elson, and W. W. Webb “Thermodynamic fluctuations in a reacting system - measurement by fluorescence correlation spectroscopy,“ Phys. Rev. Lett. |

2. | E. L. Elson and D. Magde “Fluorescence Corelation Spectroscopy I. Conceptual Basis and Theory,” Bioploymers |

3. | D. Magde, E. Elson, and W. W. Webb “Fluorescence Corelation Spectroscopy II. An Experimental Realization,” Biopolymers |

4. | J. Widengren and Ü. Mets |

5. | R. Rigler and E. Elson, Eds. |

6. | A. Benda, M. Benes, V. Marecek, A. Lhotsky, W.T. Hermens, and M. Hof, “How To Determine Diffusion Coefficients in Planar Phospholipid Systems by Confocal Fluorescence Correlation Spectroscopy,” Langmuir |

7. | J. Humpolicková, E. Gielen, A. Benda, V. Fagulova, J. Vercammen, M. VandeVen, M. Hof, M. Ameloot, and Y. Engelborghs, “Probing Diffusion Laws within Cellular Membranes by Z-Scan Fluorescence Correlation Spectroscopy,” Biophys. J. Biophys. Lett. |

8. | J. Ries and P. Schwille, “Studying Slow Membrane Dynamics with Continuous Wave Scanning Fluorescence Correlation Spectroscopy,” Biophys. J. |

9. | Z. Petrasek and P. Schwille, “Precise measurement of diffusion coefficients using scanning fluorescence correlation spectroscopy,” Biophys. J. |

10. | T. Dertinger, V. Pacheco, I. von der Hocht, R. Hartmann, I. Gregor, and J. Enderlein, “Two-focus fluorescence correlation spectroscopy: A new tool for accurate and absolute diffusion measurements,” ChemPhysChem |

11. | A. Loman, T. Dertinger, F. Koberling, and J. Enderlein, “Comparison of optical saturation effects in conventional and dual-focus fluorescence correlation spectroscopy,” Chem. Phys. Lett. |

12. | M. Böhmer, F. Pampaloni, M. Wahl, H. J. Rahn, R. Erdmann, and J. Enderlein, “Time-resolved confocal scanning device for ultrasensitive fluorescence detection,” Rev. Sci. Instrum . |

13. | B. K. Müller, E. Zaychikov, C. Bräuchle, and D. C. Lamb, “Pulsed interleaved excitation,” Biophys. J. |

14. | G. Nomarski, “Interference Microscopy - State of Art and Its Future,” J. Opt. Soc. Am. |

15. | E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. Roy. Soc. London A |

16. | B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. London A |

17. | P. R. T. Munro and P. Török, “Vectorial, high numerical aperture study of Nomarski's differential interference contrast microscope,” Opt. Express |

18. | P. Török, Z. Varga, G. R. Laczik, and J. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A |

19. | P. Török and P. Varga, “Electromagnetic diffraction of light focused through a stratified medium,” Appl. Opt. |

20. | A. Egner, M. Schrader, and S. W. Hell, “Refractive index mismatch induced intensity and phase variations in fluorescence confocal, multiphoton and 4Pi-microscopy,” Opt. Commun. |

21. | O. Haeberlé, “Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part II: confocal and multiphoton microscopy,” Opt. Commun. |

22. | O. Haeberlé, M. Ammar, H. Furukawa, K. Tenjimbayashi, and P. Török, “The point spread function of optical microscopes imaging through stratified media,” Opt. Express |

23. | I. Gregor and J. Enderlein “Focusing astigmatic Gaussian beams through optical systems with a high numerical aperture,” Opt. Lett. |

24. | J. Enderlein, I. Gregor, D. Patra, T. Dertinger, and U. B. Kaupp, “Performance of Fluorescence Correlation Spectroscopy for Measuring Diffusion and Concentration,” ChemPhysChem |

25. | M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express |

26. | I. Gregor, D. Patra, and J. Enderlein, “Optical Saturation in Fluorescence Correlation Spectroscopy under Continuous-Wave and Pulsed Excitation,” ChemPhysChem |

27. | P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. |

28. | P. D. Higdon, P. Török, and T. Wilson, “Imaging properties of high aperture multiphoton fluorescence scanning optical microscopes,” J. Microsc. |

29. | P. Török, “Propagation of electromagnetic dipole waves through dielectric interfaces,” Opt. Lett. |

30. | M. Leutenegger and T. Lasser, “Detection efficiency in total internal reflection fluorescence microscopy,” Opt. Express |

31. | M. Wahl, I. Gregor, M. Patting, and J. Enderlein, “Fast calculation of fluorescence correlation data with asynchronous time-correlated single-photon counting,” Opt. Express |

32. | C. B. Müller, K. Weiß, W. Richtering, A. Loman, and J. Enderlein, “Calibrating Differential Interference Contrast Microscopy with dual-focus Fluorescence Correlation Spectroscopy,” Opt. Express |

33. | C. B. Müller, A. Loman, V. Pacheco, F. Koberling, D. Willbold, W. Richtering, and J. Enderlein, “Precise Measurement of Diffusion by Multi-Color Dual-Focus Fluorescence Correlation Spectroscopy,” Eur. Phys. Lett. |

34. | G. Donnert, C. Eggeling, and S. W Hell, “Major signal increase in fluorescence microscopy through dark-state relaxation,” Nature Meth. |

35. | G. Nishimura and M. Kinjo “Systematic error in fluorescence correlation measurements identified by a simple saturation model of fluorescence,” Anal. Chem. |

36. | K. Berland and G. Shen, “Excitation Saturation in Two-Photon Fluorescence Correlation Spectroscopy,” Appl. Opt. |

37. | I. Gregor, D. Patra, and J. Enderlein, “Optical Saturation in Fluorescence Correlation Spectroscopy under Continuous-Wave and Pulsed Excitation,” ChemPhysChem |

**OCIS Codes**

(170.6280) Medical optics and biotechnology : Spectroscopy, fluorescence and luminescence

(180.1790) Microscopy : Confocal microscopy

(300.2530) Spectroscopy : Fluorescence, laser-induced

**ToC Category:**

Spectroscopy

**History**

Original Manuscript: July 3, 2008

Revised Manuscript: August 12, 2008

Manuscript Accepted: August 13, 2008

Published: August 29, 2008

**Virtual Issues**

Vol. 3, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

Thomas Dertinger, Anastasia Loman, Benjamin Ewers, Claus B. Müller, Benedikt Krämer, and Jörg Enderlein, "The optics and performance of dual-focus fluorescence correlation spectroscopy," Opt. Express **16**, 14353-14368 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-19-14353

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### References

- D. Magde, E. Elson, and W. W. Webb "Thermodynamic fluctuations in a reacting system - measurement by fluorescence correlation spectroscopy," Phys. Rev. Lett. 29, 705-708, (1972). [CrossRef]
- E. L. Elson and D. Magde "Fluorescence Corelation Spectroscopy I. Conceptual Basis and Theory," Bioploymers 13, 1-27 (1974). [CrossRef]
- D. Magde, E. Elson, and W. W. Webb "Fluorescence Corelation Spectroscopy II. An Experimental Realization," Biopolymers 13, 29-61 (1974). [CrossRef] [PubMed]
- J. Widengren and ??. Mets, Single-Molecule Detection in Solution - Methods and Applications, Eds. C. Zander, J. Enderlein, and R. A. Keller (Wiley-VCH, 2002) pp. 69-95. [CrossRef]
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