## Measuring diffusion with polarization-modulation dual-focus fluorescence correlation spectroscopy

Optics Express, Vol. 16, Issue 19, pp. 14609-14616 (2008)

http://dx.doi.org/10.1364/OE.16.014609

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### Abstract

We present a new technique, polarization-modulation dual-focus fluorescence correlation spectroscopy (pmFCS), based on the recently introduced dual-focus fluorescence correlation spectroscopy (2fFCS) to measure the absolute value of diffusion coefficients of fluorescent molecules at pico- to nanomolar concentrations. Analogous to 2fFCS, the new technique is robust against optical saturation in yielding correct values of the diffusion coefficient. This is in stark contrast to conventional FCS where optical saturation leads to an apparent decrease in the determined diffusion coefficient with increasing excitation power. However, compared to 2fFCS, the new technique is simpler to implement into a conventional confocal microscope setup and is compatible with cw-excitation, only needing as add-ons an electro-optical modulator and a differential interference contrast prism. With pmFCS, the measured diffusion coefficient (*D*) for Atto655 maleimide in water at 25oC is determined to be equal to (4.09±0.07)×10^{-6}cm^{2}/s, in good agreement with the value of 4.04×10^{-6}cm^{2}/s as measured by 2fFCS.

© 2008 Optical Society of America

## 1. Introduction

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]

*precisely know distance*between them. Instead of measuring only the ACF of the signal originating from a single focus, one measures the ACF of the signal originating from each focus, and the cross-correlation function (CCF) of the signals from both foci. By globally fitting all three curves and knowing the exact distance between foci, it is possible to extract a correct value of the diffusion coefficient even in the presence of considerable optical aberrations or optical saturation of the fluorescence. The achievable accuracy of the method was shown to be better than 5 % in absolute value.

## 2. Theory and data analysis

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, and the functions κ(

*z*) and

*w*(

*z*) are given by

*is the excitation wavelength, and 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

*0 are two (generally unknown) model parameters.*

_{R}**x**̂ is a unit vector along the

*x*-axis perpendicular to the optical axis (

*z*) connecting the center of the two focus positions, and the ε

_{1}and ε

_{2}are overall excitation and detection efficiency factors. When performing an FCS measurement with such a spatially modulate MDF, the resulting ACF can be calculated by:

*g*

_{∞}is the infinite lag time constant offset of the ACF,

*c*is the concentration of fluorescing molecules, T the modulation period (

*T*=2π/ω),

*D*the diffusion coefficient, and

*t*the lag time of the ACF. The integrations over time

*t*

_{0}and over the transversal coordinates (

*x*,

*y*) can be carried out analytically, leading to the result:

*g*and

_{ACF}*g*are given by the general expression:

_{CCF}*g*(

_{ACF}*t*)=

*g̃*(

*t*,0) and

*g*(

_{CCF}*t*)=

*g̃*(

*t*,δ). Thus, the resulting ACF, Eq.(7), is a mixing of the functions

*g*and

_{ACF}*g*with the fast modulation terms 1±cos(ωt)/2.

_{CCF}*D*, the MDF waist parameter

*w*

_{0}, and the confocal pinhole parameter

*R*

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

*a priori*. As was shown in Ref.[13

13. C. B. Müller, K. Weiß, W. Richtering, A. Loman, and J. Enderlein, “Calibrating Differential Interference Contrast Microscopy with dual-focus Fluorescence Correlation Spectroscopy,” Opt. Expr. **16**, 4322–4329 (2008). [CrossRef]

## 3. Experiment

### 3.1 Materials

### 3.2 Instruments

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]

## 3. Results and discussion

*D*) using pmFCS, the precise distance between two foci (δ) needs to be known. In practice, the best way to determine the precise value of δ is to perform a 2fFCS measurement on a reference dye with precisely known

*D*. Here we chose Atto655 maleimide as the reference dye that has a diffusion coefficient of

*D*=4.04×10

^{-6}cm

^{2}/s (Thomas Dertinger, private communication). The corresponding distance between the laser foci was fitted as 435 nm.

*g*and

_{ACF}*g*, and the fast modulation term adopts the form 1±cos(ω

_{CCF}*t*)/2, where ω is the modulation frequency (see Theory and Data Analysis).

14. K. Schätzel, *Single Photon Correlation Techniques. Dynamic Light Scattering: The method and some applications* (Clarendon Press, Oxford, 1993). [PubMed]

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

*D*upon increasing laser excitation power predicted by saturation effects, while

*D*stays virtually constant in both 2fFCS and pmFCS experiments over the range of excitation powers used. The average of measured

*D*for Atto655 maleimide in water corrected for 25oC equals to (4.09±0.07)×10

^{-6}cm

^{2}/s using pmFCS, and (4.06±0.09)×10

^{-6}cm

^{2}/s using 2fFCS. These two values agree with each other within the measurement error (standard deviation), and both agree reasonably well with the reference value with <2% error. It should be mentioned that no photobleaching was observed over the range of excitation power used here, whether in 2fFCS or in pmFCS experiments, which would have become visible as an increasing

*D*with increased excitation intensity. Compared to ref.[10

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

## 4. Conclusion

## Acknowledgement

## References

1. | A. Einstein, |

2. | D. Magde, E. Elson, and W. W. Webb, “Thermodynamic Fluctuations in a Reacting Systems - Measurement by Fluorescence Correlation Spectroscopy,” Phys. Rev. Lett. |

3. | D. M. Elliot and L. Elson, “Fluorescence correlation spectroscopy. I. Conceptual basis and theory,” Biopolymers |

4. | D. Magde, E. L. Elson, and W. W. Webb, “Fluorescence correlation spectroscopy. II. An experimental realization,” Biopolymers |

5. | J. Enderlein, I. Gregor, D. Patra, and J. Fitter, “Art and artefacts of fluorescence correlation spectroscopy,” Curr. Pharm. Biotechnol. |

6. | J. Enderlein, I. Gregor, D. Patra, T. Dertinger, and U. B. Kaupp, “Performance of fluorescence correlation spectroscopy for measuring diffusion and concentration,” ChemPhysChem |

7. | K. Berland and G. Shen, “Excitation saturation in two-photon fluorescence correlation spectroscopy,” Appl. Opt. |

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

9. | I. Gregor, D. Patra, and J. Enderlein, “Optical saturation in fluorescence correlation spectroscopy under continuous-wave and pulsed excitation,” ChemPhysChem |

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

12. | D. V. O’Connor and D. Phillips, |

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

14. | K. Schätzel, |

**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: April 10, 2008

Revised Manuscript: July 18, 2008

Manuscript Accepted: August 5, 2008

Published: September 3, 2008

**Virtual Issues**

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

**Citation**

You Korlann, Thomas Dertinger, Xavier Michalet, Shimon Weiss, and Jörg Enderlein, "Measuring diffusion with polarization-modulation dual-focus fluorescence correlation spectroscopy," Opt. Express **16**, 14609-14616 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-19-14609

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### References

- A. Einstein, Investigations on the theory of the Brownian movement (Dover Publications, New York, 1956), p. 119 p.
- D. Magde, E. Elson, and W. W. Webb, "Thermodynamic Fluctuations in a Reacting Systems - Measurement by Fluorescence Correlation Spectroscopy," Phys. Rev. Lett. 29, 705 (1972). [CrossRef]
- D. M. Elliot L. Elson, "Fluorescence correlation spectroscopy. I. Conceptual basis and theory," Biopolymers 13, 1-27 (1974). [CrossRef]
- D. Magde, E. L. Elson, and W. W. Webb, "Fluorescence correlation spectroscopy. II. An experimental realization," Biopolymers 13, 29-61 (1974). [CrossRef] [PubMed]
- J. Enderlein, I. Gregor, D. Patra, and J. Fitter, "Art and artefacts of fluorescence correlation spectroscopy," Curr. Pharm. Biotechnol. 5, 155-161 (2004). [CrossRef] [PubMed]
- 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]
- K. Berland and G. Shen, "Excitation saturation in two-photon fluorescence correlation spectroscopy," Appl. Opt. 42, 5566-5576 (2003). [CrossRef] [PubMed]
- G. Nishimura and M. Kinjo, "Systematic error in fluorescence correlation measurements identified by a simple saturation model of fluorescence," Anal. Chem. 76, 1963-1970 (2004). [CrossRef] [PubMed]
- I. Gregor, D. Patra, and J. Enderlein, "Optical saturation in fluorescence correlation spectroscopy under continuous-wave and pulsed excitation," ChemPhysChem 6, 164-170 (2005). [CrossRef] [PubMed]
- 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]
- B. K. Müller, E. Zaychikov, C. Bräuchle, and D. C. Lamb, "Pulsed interleaved excitation," Biophys. J. 89, 3508-3522 (2005). [CrossRef] [PubMed]
- D. V. O'Connor and D. Phillips, Time-correlated single photon counting (Academic Press, London; Orlando, 1984), pp. viii, 288 p.
- C. B. Müller, K. Weiβ, W. Richtering, A. Loman, and J. Enderlein, "Calibrating Differential Interference Contrast Microscopy with dual-focus Fluorescence Correlation Spectroscopy," Opt. Expr. 16, 4322-4329 (2008). [CrossRef]
- K. Schätzel, Single Photon Correlation Techniques. Dynamic Light Scattering: The method and some applications (Clarendon Press, Oxford, 1993). [PubMed]

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