## Quantifying Noise in Optical Tweezers by Allan Variance

Optics Express, Vol. 17, Issue 15, pp. 13255-13269 (2009)

http://dx.doi.org/10.1364/OE.17.013255

Acrobat PDF (1116 KB)

### Abstract

Much effort is put into minimizing noise in optical tweezers experiments because noise and drift can mask fundamental behaviours of, e.g., single molecule assays. Various initiatives have been taken to reduce or eliminate noise but it has been difficult to quantify their effect. We propose to use Allan variance as a simple and efficient method to quantify noise in optical tweezers setups. We apply the method to determine the optimal measurement time, frequency, and detection scheme, and quantify the effect of acoustic noise in the lab. The method can also be used on-the-fly for determining optimal parameters of running experiments.

© 2009 Optical Society of America

## 1. Introduction

1. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. **75**, 2787–2809 (2004). [CrossRef]

2. A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature **330**, 769–771 (1987). [CrossRef] [PubMed]

3. P. Hansen, V. Bhatia, N. Harrit, and L. B. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. **5**, 1937–1942 (2005). [CrossRef] [PubMed]

4. L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient Optical Trapping and Visualization of Silver Nanoparticles,” Nano Lett. **8**, 1486–1491 (2008). [CrossRef] [PubMed]

5. C. Selhuber-Unkel, I. Zins, O. Schubert, C. Sönnichsen, and L. B. Oddershede, “Quantitative optical trapping of single gold nanorods,” Nano Lett. **8**, 2998–3003 (2008). [CrossRef] [PubMed]

6. L. Jauffred, A. C. Richardson, and L. B. Oddershede, “Three-Dimensional Optical Control of Individual Quantum Dots,” Nano Lett. **8**, 3376–3380 (2008). [CrossRef] [PubMed]

7. E. A. Abbondanzieri, W. J. Greenleaf, J. W. Shaevitz, R. Block, and S. M. Landick, “Direct observation of base-pair stepping by RNA polymerase,” Nature **438**, 460–465 (2005). [CrossRef] [PubMed]

8. J. Liphardt, B. Onoa, S. B. Smith, I. Tinoco, and C. Bustamante, “Reversible unfolding of single RNA molecules by mechanical force,” Science **292**, 733–737 (2001). [CrossRef] [PubMed]

9. A. R. Carter, Y. Seol, and T. T. Perkins, “Precision surface-coupled optical-trapping assay with one-basepair resolution,” Biophys. J. **96**, 2926–2934 (2009). [CrossRef] [PubMed]

10. F. Gittes and C. F. Schmidt, “Signals and noise in micromechanical measurements,” Methods Cell. Biol. **55**, 129–156 (1998). [CrossRef]

11. M. Klein, M. Andersson, O. Axner, and E. Fallman, “Dual-trap technique for reduction of low-frequency noise in force measuring optical tweezers,” Appl. Opt. **46**, 405–412 (2007). [CrossRef] [PubMed]

12. D. W. Allan, “Statistics of atomic frequency standards,” Proc. IEEE **54**, 221–230 (1966). [CrossRef]

14. G. M. Gibson, J. Leach, S. Keen, A. J. Wright, and M. J. Padgett, “Measuring the accuracy of particle position and force in optical tweezers using high-speed video microscopy,” Opt. Express **16**, 14561–14570 (2008). [CrossRef] [PubMed]

## 2. Materials and Methods

*µ*m. Spheres were trapped 40

*µ*m above the lower glass slip to avoid significant interaction with adjacent surfaces [10

10. F. Gittes and C. F. Schmidt, “Signals and noise in micromechanical measurements,” Methods Cell. Biol. **55**, 129–156 (1998). [CrossRef]

^{24}consecutive points. For an acquisition rate of

*f*

_{acq}=22 kHz, corresponding to time series of about 13 min. Apart from filters inherently present in the detection diodes and electronics [16

16. K. Oddershede, L. B. Berg-Sørensen, E. L. Florin, and H. Flyvbjerg, “Unintended filtering in a typical photodiode detection system for optical tweezers,” J. Appl. Phys. **93**, 3167–3176 (2003). [CrossRef]

### 2.1. Calibration

*F*=-κ

*x*, where

*κ*is denoted the trap stiffness and

*x*is the deviation from the object’s equilibrium position. The goal of the calibration procedure is to extract

*κ*which completely characterizes the thermal motion of the trapped object. The equation of motion is well described by the Langevin equation, and a Fourier transformation yields the positional power spectrum, from which the ratio between the trap stiffness and the friction coefficient γ can be found [10

10. F. Gittes and C. F. Schmidt, “Signals and noise in micromechanical measurements,” Methods Cell. Biol. **55**, 129–156 (1998). [CrossRef]

*f*:

_{c}*γ*is given by Stokes law: γ=6

*πηr*, where

*η*the viscosity of water (

*η*=8.9·10

^{-4}sPa), and

*r*the radius of the sphere. The power spectral analysis also provides information regarding the conversion factor

*β*which converts the position measured in volts by the photodiodes to position travelled in nanometers by the trapped sphere [17].

*f*

_{acq}=22 kHz, we used 2

^{16}data points for one individual calibration. Calibration was performed by the power-spectrum method as described in [18

18. K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. **75**, 594–612 (2004). [CrossRef]

19. P. M. Hansen, I. Toliç-Nørrelykke, H. Flyvbjerg, and K. Berg-Sørensen, “*tweezercalib 2.1*: Faster version of MATLAB package for precise calibration of optical tweezers,” Comput. Phys. Commun.175, 572–573 (2006). [CrossRef]

11. M. Klein, M. Andersson, O. Axner, and E. Fallman, “Dual-trap technique for reduction of low-frequency noise in force measuring optical tweezers,” Appl. Opt. **46**, 405–412 (2007). [CrossRef] [PubMed]

9. A. R. Carter, Y. Seol, and T. T. Perkins, “Precision surface-coupled optical-trapping assay with one-basepair resolution,” Biophys. J. **96**, 2926–2934 (2009). [CrossRef] [PubMed]

### 2.2. Monte-Carlo Simulations

20. F. Czerwinski, “BeadFluct v1.0,” MatlabCentral **24196** (2009), http://www.mathworks.com/matlabcentral/fileexchange/24196.

^{-16}. The radius of the sphere

*r*, the corner frequency

*f*, the trapping stiffness κ, the acquisition frequency

_{c}*f*

_{acq}, and the number of sampled points

*N*could be varied. Adjacent time series gained by simulations were treated as regular data and compared to experimental results.

### 2.3. Allan Variance

*Definition*. Given a time series of

*N*elements and a total measurement time of

*t*

_{acq}=

*f*

_{acq}

*N*, then Allan variance is defined as:

*x*being the mean over a time interval of a length

_{i}*τ*=

*f*

_{acq}

*m, m*the number of elements in this particular interval. 〈…〉 denotes the arithmetric mean. In words, the Allan variance is half the averaged squared mean of neighboring intervals. Consequently, Allan variance can only be calculated for

*τ*≤

*t*

_{acq}/2, and it is always ≥0. Using

*x*̂=

*βx*, the linearity of the Allan variance follows directly from Eq. (2):

*τ*≪

*t*

_{acq}. For

*τ*<tacq, SE

*(*

_{σ}*τ*) can be approximated by utilizing

*n*≈⌊

*N/m*⌋=⌊

*t*

_{acq}/τ

*⌋*. Note, in this definition the neighboring time intervals of length

*τ*are in principle conditionally independent. When

*τ*is on the order of tacq one faces the problem of low statistics.

*Overlapping Allan Variance*. To account for the huge statistical error of the Allan variance, Eq. (4), in case of large

*τ*, one can give up the statistically independence of time intervals and use a sliding interval instead. One calculates the difference of two neighboring intervals each containing

*m*elements. Then the intervals move one element further in time while both intervals loose their ‘oldest’ element, conserving their number of elements. Consequently, the statistical error scales with

*n*as in Eq. (4), the conditionally independent case.

*Normal variance*. Often, the standard way to calculate variance as the square of the standard deviation of the mean with a sliding window averaging over τ has been used to evaluate drift in a data set with mean

*x*̄:

### 2.4. Thermal Limit

*x*

^{2}〉=2

*Dτ*with the diffusion constant

*D*=

*k*. It can be further confined by a trapping potential characterized by a trapping constant

_{a}T/γ*κ*. Using the same approximation as for Eq. (4), the statistical error for a trapped sphere become

11. M. Klein, M. Andersson, O. Axner, and E. Fallman, “Dual-trap technique for reduction of low-frequency noise in force measuring optical tweezers,” Appl. Opt. **46**, 405–412 (2007). [CrossRef] [PubMed]

### 2.5. Allan Variance for Optical Tweezers Setups

*τ*, one of the requirements is a device which allows for an acquisition of positional data at a fast and a reliable rate over a (very) long measurement time.

22. F. Czerwinski, “allan v1.71,” MatlabCentral **21727** (2008), http://www.mathworks.com/matlabcentral/fileexchange/21727.

*f*

_{acq}=22 kHz for about 13 min, i.e. 2

^{24}data points for each coordinate). Then we altered the parameters and repeated the measurements. Though the Figs. of the results section represent plots of particular datasets, all experiments have been repeated as described above and all results explicitly stated are solid and reproducible. We found it advantageous to plot Allan variances in log-log plots. In those, slopes correspond to the exponent of a proposed relation.

## 3. Results

*N*=262,144 which overlap precisely. In (c) the positional power spectral density of the two data sets are plotted, the full lines show Lorentzian fits to the power spectra. These fits include the effect of aliasing and for the experimental data the filtering effect of the photodetection system and electronics as implemented in the software [19

19. P. M. Hansen, I. Toliç-Nørrelykke, H. Flyvbjerg, and K. Berg-Sørensen, “*tweezercalib 2.1*: Faster version of MATLAB package for precise calibration of optical tweezers,” Comput. Phys. Commun.175, 572–573 (2006). [CrossRef]

*f*[18

_{c}18. K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. **75**, 594–612 (2004). [CrossRef]

*κ*(Eq. (1)). At high frequencies one sees that the well understood effect of photodiode filtering [16

16. K. Oddershede, L. B. Berg-Sørensen, E. L. Florin, and H. Flyvbjerg, “Unintended filtering in a typical photodiode detection system for optical tweezers,” J. Appl. Phys. **93**, 3167–3176 (2003). [CrossRef]

18. K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. **75**, 594–612 (2004). [CrossRef]

### 3.1. Allan Variance for Individual Experiments

*κ*=33.6 pN/

*µ*m. The green-grey curves stem from a Monte-Carlo simulation. The orange curves are from an experiment with a stronger trap,

*f*=1600 Hz, κ=67.7 pN/

_{c}*µ*m. The orange-grey curves are from the corresponding simulation. The full lines denote the overlapping Allan variance. Due to its improved statistics, it is always inside the variation of the Allan variance. This is particularly evident for large τ.

### 3.2. Accuracy Depends on Acquisition Frequency

*f*

_{acq}. A polystyrene sphere was optically trapped and the data acquisition frequency was varied between 10 Hz and 100 kHz. The result of the corresponding overlapping Allan variance is shown in Fig. 4.

*τ*

^{-1}

*and depends on the trapping stiffness. It can be found in the plotted data set at*

_{c}*τ*

^{-1}

^{c}=2

*πf*≈10 kHz. Above this threshold, an increase in frequency does not change the Allan variance for the majority of the measurement times. Only above the optimal interval, around 10 s, it seems beneficial to improve the acquisition frequency above 10 kHz. However, the bandwidth of the used position-sensitive photodiode was found to lie in between 43 and 47 kHz. Hence, the apparent aliasing might be the reason for the steeper slope for measurement times

_{c}*τ*>20 s in the plotted graphs.

*f*acq<τ

^{-1}

*one measures thermal fluctuations of the sphere without any informational content.*

_{c}### 3.3. Acoustics and Chamber Geometry

*x*), 1 cm (

*y*), and 95

*µ*m (

*z*) and the sphere was trapped approximately in the center of this chamber. The green traces in Fig. 5 are Allan variances of the adjacent time series in

*x*, the red traces those of

*y*. The black full lines denote the thermal limit. The measurement chamber was sealed with a thin glass cover slip at the bottom towards the focusing objective and at the top either a thick cover slip (1 mm, data in Fig. 5(a)) or a thin cover slip (0.13–0.16 mm, data in Fig. 5(b)).

*y*-direction, the smallest lateral dimension of the chamber. However, in the

*x*-direction, the longest lateral direction of the chamber, there is a significant effect of reducing acoustic noise. To check the directional dependence of these results we rotated the chamber 90° and still saw the effect. In other words, the difference in

*x*and

*y*directions are due to sample geometry and not to asymmetries in the laser or detection system. All other plots of Allan variances in the present letter originate from the longest direction in an elongated chamber (corresponding to the

*x*-direction in Fig. 5).

*τ*>20 s.

### 3.4. Piezo Stage

*κ*=33.6 pN/

*µ*m (violet) with the piezo on. When decreasing the trap stiffness to

*κ*=15.3 pN/

*µ*m (red), the loss in accuracy over measurement intervals of 0.5 s<

*τ*<110 s became even more pronounced; whereas switching off the piezo and keeping the trap stiffness constant at

*κ*=15.3 pN/

*µ*m (green) eliminated the effect. Hence, the piezo does add noise to the system, in particular in a measurement time interval 0.5 s<

*τ*<110 s. Furthermore, Fig. 6 confirms the findings from Fig. 3 that stronger traps can partially eliminate low-frequency noise. The black trace shown in Fig. 6 is the Allan variance of the output of the control box yielding the position of the piezo while the sample was mounted. For short measurement times, the Allan variance shows local maxima that correspond to odd-numbered divisors of the piezo’s resonance frequency. The resonance of the piezo has thus not been altered significantly by mounting the stage. At measurement times within 0.5 s<

*τ*<110 s, the noise of the stage alone peaks in the same interval as the noise peak in the violet and red curves. Therefore, we propose that the piezo stage contributes to the noise as quantified by Allan variance in a frequency interval as visualized by the grey shading in Fig. 6. Those low-frequency phenomena would be challenging to identify using standard noise-detection methods.

### 3.5. Comparison of Photodiodes

*κ*=33.6 pN/

*µm*, using either the position-sensitive diode (PSD, green) or the quadrant photodiode (QPD, orange). It was the same sphere trapped in both cases. The performances of the two types of photodiodes were quite similar, but for the shortest as well as for the longest measurement times the PSD had a smaller Allan variance. Another experiment was performed where there was nothing inside the optical trap, but the laser was shining directly at the photo detection system. An optically trapped sphere has a focusing effect on the laser light, so in the absence of a trapped sphere the total signal on the detection system is significantly smaller than when a sphere is trapped using the same laser intensity. The dashed lines in Fig. 7 show the Allan variance of the laser signal as detected by the QPD (orange) and PSD (green), respectively. Generally, the noise of the PSD is lower than of the QPD. Interestingly, one sees a pronounced peak in the Allan variance of the QPD around 16 ms. This time interval corresponds exactly to the detection delay time reported for this particular diode [16

16. K. Oddershede, L. B. Berg-Sørensen, E. L. Florin, and H. Flyvbjerg, “Unintended filtering in a typical photodiode detection system for optical tweezers,” J. Appl. Phys. **93**, 3167–3176 (2003). [CrossRef]

## 4. Discussion

*µ*m are used, the optimal length of a time series for calibration is in the order of seconds, not e.g. tens of seconds. This time scale is on the same order as reported in [14

14. G. M. Gibson, J. Leach, S. Keen, A. J. Wright, and M. J. Padgett, “Measuring the accuracy of particle position and force in optical tweezers using high-speed video microscopy,” Opt. Express **16**, 14561–14570 (2008). [CrossRef] [PubMed]

14. G. M. Gibson, J. Leach, S. Keen, A. J. Wright, and M. J. Padgett, “Measuring the accuracy of particle position and force in optical tweezers using high-speed video microscopy,” Opt. Express **16**, 14561–14570 (2008). [CrossRef] [PubMed]

**16**, 14561–14570 (2008). [CrossRef] [PubMed]

**93**, 3167–3176 (2003). [CrossRef]

**16**, 14561–14570 (2008). [CrossRef] [PubMed]

## 5. Conclusions

## Acknowledgements

## References and links

1. | K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. |

2. | A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature |

3. | P. Hansen, V. Bhatia, N. Harrit, and L. B. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. |

4. | L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient Optical Trapping and Visualization of Silver Nanoparticles,” Nano Lett. |

5. | C. Selhuber-Unkel, I. Zins, O. Schubert, C. Sönnichsen, and L. B. Oddershede, “Quantitative optical trapping of single gold nanorods,” Nano Lett. |

6. | L. Jauffred, A. C. Richardson, and L. B. Oddershede, “Three-Dimensional Optical Control of Individual Quantum Dots,” Nano Lett. |

7. | E. A. Abbondanzieri, W. J. Greenleaf, J. W. Shaevitz, R. Block, and S. M. Landick, “Direct observation of base-pair stepping by RNA polymerase,” Nature |

8. | J. Liphardt, B. Onoa, S. B. Smith, I. Tinoco, and C. Bustamante, “Reversible unfolding of single RNA molecules by mechanical force,” Science |

9. | A. R. Carter, Y. Seol, and T. T. Perkins, “Precision surface-coupled optical-trapping assay with one-basepair resolution,” Biophys. J. |

10. | F. Gittes and C. F. Schmidt, “Signals and noise in micromechanical measurements,” Methods Cell. Biol. |

11. | M. Klein, M. Andersson, O. Axner, and E. Fallman, “Dual-trap technique for reduction of low-frequency noise in force measuring optical tweezers,” Appl. Opt. |

12. | D. W. Allan, “Statistics of atomic frequency standards,” Proc. IEEE |

13. | P. Banerjee, A. Chatterjee, and A. Suman, “Determination of Allan deviation of Cesium atomic clock for lower averaging time,” Indian J. Pure Appl. Phys. |

14. | G. M. Gibson, J. Leach, S. Keen, A. J. Wright, and M. J. Padgett, “Measuring the accuracy of particle position and force in optical tweezers using high-speed video microscopy,” Opt. Express |

15. | F. Czerwinski and L. B. Oddershede, “Reliable Data-Streaming Software for Photodiode Readout in LABVIEW,” in. prep. (2009). |

16. | K. Oddershede, L. B. Berg-Sørensen, E. L. Florin, and H. Flyvbjerg, “Unintended filtering in a typical photodiode detection system for optical tweezers,” J. Appl. Phys. |

17. | L. B. Oddershede, S. Grego, S. Nørrelykke, and K. Berg-Sørensen, “Optical tweezers: probing biological surfaces,” Probe Microsc |

18. | K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. |

19. | P. M. Hansen, I. Toliç-Nørrelykke, H. Flyvbjerg, and K. Berg-Sørensen, “ |

20. | F. Czerwinski, “BeadFluct v1.0,” MatlabCentral |

21. | P. Kartaschoff, |

22. | F. Czerwinski, “allan v1.71,” MatlabCentral |

**OCIS Codes**

(000.3110) General : Instruments, apparatus, and components common to the sciences

(000.5490) General : Probability theory, stochastic processes, and statistics

(030.4280) Coherence and statistical optics : Noise in imaging systems

(110.4280) Imaging systems : Noise in imaging systems

(140.7010) Lasers and laser optics : Laser trapping

(350.4800) Other areas of optics : Optical standards and testing

**ToC Category:**

Optical Tweezers or Optical Manipulation

**History**

Original Manuscript: July 1, 2009

Revised Manuscript: July 15, 2009

Manuscript Accepted: July 15, 2009

Published: July 17, 2009

**Virtual Issues**

Vol. 4, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Fabian Czerwinski, Andrew C. Richardson, and Lene B. Oddershede, "Quantifying Noise in Optical Tweezers by Allan Variance," Opt. Express **17**, 13255-13269 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-13255

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### References

- K. C. Neuman, and S. M. Block, "Optical trapping," Rev. Sci. Instrum. 75, 2787-2809 (2004). [CrossRef]
- A. Ashkin, J. M. Dziedzic, and T. Yamane, "Optical trapping and manipulation of single cells using infrared laser beams," Nature 330, 769-771 (1987). [CrossRef] [PubMed]
- P. Hansen, V. Bhatia, N. Harrit, and L. B. Oddershede, "Expanding the optical trapping range of gold nanoparticles," Nano Lett. 5, 1937-1942 (2005). [CrossRef] [PubMed]
- L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, "Efficient Optical Trapping and Visualization of Silver Nanoparticles," Nano Lett. 8, 1486-1491 (2008). [CrossRef] [PubMed]
- C. Selhuber-Unkel, I. Zins, O. Schubert, C. S¨onnichsen, and L. B. Oddershede, "Quantitative optical trapping of single gold nanorods," Nano Lett. 8, 2998-3003 (2008). [CrossRef] [PubMed]
- L. Jauffred, A. C. Richardson, and L. B. Oddershede, "Three-Dimensional Optical Control of Individual Quantum Dots," Nano Lett. 8, 3376-3380 (2008). [CrossRef] [PubMed]
- E. A. Abbondanzieri, W. J. Greenleaf, J. W. Shaevitz, R. Landick, and S. M. Block, "Direct observation of base-pair stepping by RNA polymerase," Nature 438, 460-465 (2005). [CrossRef] [PubMed]
- J. Liphardt, B. Onoa, S. B. Smith, I. Tinoco, and C. Bustamante, "Reversible unfolding of single RNA molecules by mechanical force," Science 292, 733-737 (2001). [CrossRef] [PubMed]
- A. R. Carter, Y. Seol, and T. T. Perkins, "Precision surface-coupled optical-trapping assay with one-basepair resolution," Biophys. J. 96, 2926-2934 (2009). [CrossRef] [PubMed]
- F. Gittes and C. F. Schmidt, "Signals and noise in micromechanical measurements," Methods Cell. Biol. 55, 129-156 (1998). [CrossRef]
- M. Klein, M. Andersson, O. Axner, and E. Fallman, "Dual-trap technique for reduction of low-frequency noise in force measuring optical tweezers," Appl. Opt. 46, 405-412 (2007). [CrossRef] [PubMed]
- D. W. Allan, "Statistics of atomic frequency standards," Proc. IEEE 54, 221-230 (1966). [CrossRef]
- P. Banerjee, A. Chatterjee, and A. Suman, "Determination of Allan deviation of Cesium atomic clock for lower averaging time," Indian J. Pure Appl. Phys. 45, 945-949 (2007).
- G. M. Gibson, J. Leach, S. Keen, A. J. Wright, and M. J. Padgett, "Measuring the accuracy of particle position and force in optical tweezers using high-speed video microscopy," Opt. Express 16, 14561-14570 (2008). [CrossRef] [PubMed]
- F. Czerwinski and L. B. Oddershede, "Reliable Data-Streaming Software for Photodiode Readout in LABVIEW," in. prep. (2009).
- K. Berg-Sørensen, L. B. Oddershede, E. L. Florin, and H. Flyvbjerg, "Unintended filtering in a typical photodiode detection system for optical tweezers," J. Appl. Phys. 93, 3167-3176 (2003). [CrossRef]
- L. B. Oddershede, S. Grego, S. Nørrelykke, and K. Berg-Sørensen, "Optical tweezers: probing biological surfaces," Probe Microsc 2, 129-137 (2001).
- K. Berg-Sørensen and H. Flyvbjerg, "Power spectrum analysis for optical tweezers," Rev. Sci. Instrum. 75, 594- 612 (2004). [CrossRef]
- P. M. Hansen, I. Tolic¸-Nørrelykke, H. Flyvbjerg, and K. Berg-Sørensen, "tweezercalib 2.1: Faster version of MATLAB package for precise calibration of optical tweezers," Comput. Phys. Commun. 175, 572-573 (2006). [CrossRef]
- F. Czerwinski, "BeadFluct v1.0," MatlabCentral 24196 (2009), http://www.mathworks.com/matlabcentral/fileexchange/24196.
- P. Kartaschoff, Frequency and Time (Academic Press, 1978).
- F. Czerwinski, "allan v1.71," MatlabCentral 21727 (2008), http://www.mathworks.com/matlabcentral/fileexchange/21727.

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