## Measuring the accuracy of particle position and force in optical tweezers using high-speed video microscopy

Optics Express, Vol. 16, Issue 19, pp. 14561-14570 (2008)

http://dx.doi.org/10.1364/OE.16.014561

Acrobat PDF (243 KB)

### Abstract

We assess the performance of a CMOS camera for the measurement of particle position within optical tweezers and the associated autocorrelation function and power spectrum. Measurement of the displacement of the particle from the trap center can also be related to the applied force. By considering the Allan variance of these measurements, we show that such cameras are capable of reaching the thermal limits of nanometer and femtonewton accuracies, and hence are suitable for many of the applications that traditionally use quadrant photodiodes. As an example of a multi-particle measurement we show the hydrodynamic coupling between two particles.

© 2008 Optical Society of America

## 1. Introduction

3. J.-C. Meiners and S. R. Quake, “Femtonewton force spectroscopy of single extended DNA molecules,” Phys. Rev. Lett. **84**, 5014–5017 (2000). [CrossRef] [PubMed]

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

5. M. Polin, D. G. Grier, and S. R. Quake, “Anomalous vibrational dispersion in holographically trapped colloidal arrays,” Phys. Rev. Lett. **96**, 088101 (2006). [CrossRef] [PubMed]

6. S. Keen, J. Leach, G. Gibson, and M. Padgett, “Comparison of a high-speed camera and a quadrant detector for measuring displacements in optical tweezers,” J. Opt. A: Pure Appl. Opt. **9**, S264–S266 (2007). [CrossRef]

7. O. Otto, C. Gutsche, F. Kremer, and U. F. Keyser, “Optical tweezers with 2.5kHz bandwidth video detection for single-colloid electrophoresis,” Rev. Sci. Instrum . **79**, 023710 (2008). [CrossRef] [PubMed]

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

## 2. Experimental configuration

^{2}, SolsTiS) which provides up to 1W at 830nm. The laser is expanded to slightly overfill the aperture of a spatial light modulator (SLM) (Hamamatsu, LCOS X10468-02), allowing multiple optical traps to be created, and then coupled into the tweezers system by imaging the SLM on to the back aperture of the microscope objective lens. The device is gamma corrected such that approximately 60% of the incident light is diffracted into the desired trap pattern. Using our SLM control software [9

9. J. Leach, K. Wulff, G. Sinclair, P. Jordan, J. Courtial, L. Thomson, G. Gibson, K. Karunwi, J. Cooper, Z. J. Laczik, and M. Padgett, “Interactive approach to optical tweezers control,” Appl. Opt. **45**, 897–903 (2006). [CrossRef] [PubMed]

10. G. Gibson, D. M. Carberry, G. Whyte, J. Leach, J. Courtial, J. C. Jackson, D. Robert, M. Miles, and M. Padgett, “Holographic assembly workstation for optical manipulation,” J. Opt. A: Pure Appl. Opt. **10**, 044009 (2008). [CrossRef]

*µ*m diameter silica beads and position them anywhere within the field of view. The tweezers is based around an inverted microscope, where the same objective lens, 100x 1.3NA, (Zeiss, Plan-Neofluor) is used to both focus the trapping beam and to image the resulting motion of the particles. Samples containing 2

*µ*m diameter silica beads in water are mounted in a motorized microscope stage (ASI, MS-2000). The stage allows accurate control of the sample position and provides a known displacement of a fixed particle, or bead, for calibrating the camera.

*µ*m.

## 3. Position measurements, autocorrelations and power spectra

**κ**[2

2. J. E. Molloy and M. J. Padgett, “Lights, action: optical tweezers,” Contemp. Phys. **43**, 241–258 (2002). [CrossRef]

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

*x*

^{2}〉 is given by the equipartition of energy [11

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

*x*

^{2}〉 is a common method for deducing

**κ**. Since the oscillator is significantly over damped, the autocorrelation of particle position is described by a single exponential decay of time constant

*τ*

_{0}=1/2

*πf*

_{0}, where

*f*

_{0}is the knee frequency above which the particle can be considered to be free, given by [11

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

*γ*=6

*πrη*,

*η*is the viscosity of the surrounding fluid and

*r*is the radius of the particle.

**κ**=5.6E-6 N/m) and high (

**κ**=2.3E-5 N/m) trap strengths respectively. This shows that the camera system is sufficient for measuring the thermal motion of the trapped particles, easily distinguished from the sensor noise revealed by the fixed particle. Alternatively, the thermal motion of the trapped particles is often studied by plotting the autocorrelation (Fig. 3) or power spectrum (Fig. 4). However, although sufficient for some applications, none of these results show how the sensor performance varies over differing averaging timescales, nor over which timescales the sensor reaches a performance level sufficient to record the true thermal motion of the particles.

## 4. Allan variance of position measurements

*xy*positions of trapped silica beads over a duration of 5 minutes (300,000 data points). From the

*xy*data we calculated the Allan variance of position given by

*x*is the average position over the sample period

_{n}*n*, and

*τ*is the time per sample period.

*x*

^{2}〉 and the number of independent measurements,

*N*. In a time Δ

*t*, the number of independent measurements is given by

*SE*

_{〈x〉}, to be

12. J. R. Moffitt, Y. R. Chemla, D. Izhaky, and C. Bustamante, “Differential detection of dual traps improves the spatial resolution of optical tweezers,” PNAS **103**, 9006–9011 (2006). [CrossRef] [PubMed]

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

14. M. Atakhorrami, K. M. Addas, and C. F. Schmidt, “Twin optical traps for two-particle cross-correlation measurements: Eliminating cross-talk,” Rev. Sci. Instrum. **79**, 043103 (2008). [CrossRef] [PubMed]

**σ**

^{2}

*(*

_{x}**τ**), of a 2

*µ*m silica bead for different values of optical trap stiffness. Also shown is the Allan variance for the differential position between two beads trapped 10

*µ*m apart. For comparison the stronger trapped bead is plotted in relation to the estimate for the thermally limited precision given by Eq. 5. At timescales short compared to the autocorrelation time of the trap, the bead is moving with a uniform velocity and hence the Allan variance increases with time. At timescales above the autocorrelation time, the bead positions are randomly distributed within the trap and the accuracy of the mean improves with the square root of the averaging time. We see that for a single trap the minimum error is of order of 1nm obtained for an averaging time of order 1s. Above this time we see that the Allan variance increases, which is a result of drift within the system. The longer term stability of the system can be improved by making measurements on the differential position of two beads, which effectively drift together. However, this improvement in long term stability is only at the expense of a √2 increase in noise since the Brownian motion of the two beads add in quadrature. Shown also on the graph is the Allan variance of the position measurement of beads fixed to the cover slip. At short timescales the Allan variance is limited only by the inherent measurement noise of the camera technique, but at longer timescales it increases above that of the trapped bead. This increase at long timescales is indicates that the thermal, or other, stability of the sample stage is worse than the pointing stability of the laser.

15. J.-C. Meiners and S. R. Quake, “Direct measurement of hydrodynamic cross correlations between two particles in an external potential,” Phys. Rev. Lett. **82**, 2211–2214 (1999). [CrossRef]

*µ*m beads trapped 3

*µ*m apart, the traces are averaged over 30, 2 second data sets. We see that the functions are smoothly varying, showing no evidence of underlying “digitization” even at length scales <20nm

^{2}.

## 5. Allan variance of force measurements

*F*〉, can be inferred from observation of the particle displacement from trap center, 〈

*F*〉=

**κ**(〈

*x*〉-

*x*

_{0}), where both

**κ**and

*x*

_{0}can be determined from the positional data prior to the application of the force. Using the same data as for the standard error in the particle position, the equivalent standard error in the applied force is given as

**κ**increases the precision to which the particle position can be determined, it reduces the displacement for a given force. The result is a precision of thermally limited force measurement that is independent of

**κ**.

**σ**

^{2}

*F*(

**τ**)=

**σ**

^{2}

_{x}(

**τ**)

**κ**

^{2}. The plots in Fig. 7 show the Allan variance of measurements of force acting on a 2

*µ*m silica bead for different values of

**κ**. Also shown is the Allan variance for the differential force between two beads. For comparison these are plotted in relation to the estimate for the thermally limited precision given by Eq. 6. As discussed above, we see that for timescales longer than the autocorrelation time of the trap the precision of force measurement is independent of

**κ**. At longer timescales the precision is compromised by the drift in the system. One sees that a force measurement precision of 10fN can be obtained for an averaging time of a few seconds.

## 6. Discussion and conclusions

17. R. Di Leonardo, S. Keen, J. Leach, C. D. Saunter, G. D. Love, G. Ruocco, and M. J. Padgett, “Eigenmodes of a hydrodynamically coupled micron-size multiple-particle ring,” Phys. Rev. E **76**, 061402 (2007). [CrossRef]

## Acknowledgments

## References and links

1. | A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. |

2. | J. E. Molloy and M. J. Padgett, “Lights, action: optical tweezers,” Contemp. Phys. |

3. | J.-C. Meiners and S. R. Quake, “Femtonewton force spectroscopy of single extended DNA molecules,” Phys. Rev. Lett. |

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

5. | M. Polin, D. G. Grier, and S. R. Quake, “Anomalous vibrational dispersion in holographically trapped colloidal arrays,” Phys. Rev. Lett. |

6. | S. Keen, J. Leach, G. Gibson, and M. Padgett, “Comparison of a high-speed camera and a quadrant detector for measuring displacements in optical tweezers,” J. Opt. A: Pure Appl. Opt. |

7. | O. Otto, C. Gutsche, F. Kremer, and U. F. Keyser, “Optical tweezers with 2.5kHz bandwidth video detection for single-colloid electrophoresis,” Rev. Sci. Instrum . |

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

9. | J. Leach, K. Wulff, G. Sinclair, P. Jordan, J. Courtial, L. Thomson, G. Gibson, K. Karunwi, J. Cooper, Z. J. Laczik, and M. Padgett, “Interactive approach to optical tweezers control,” Appl. Opt. |

10. | G. Gibson, D. M. Carberry, G. Whyte, J. Leach, J. Courtial, J. C. Jackson, D. Robert, M. Miles, and M. Padgett, “Holographic assembly workstation for optical manipulation,” J. Opt. A: Pure Appl. Opt. |

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

12. | J. R. Moffitt, Y. R. Chemla, D. Izhaky, and C. Bustamante, “Differential detection of dual traps improves the spatial resolution of optical tweezers,” PNAS |

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

14. | M. Atakhorrami, K. M. Addas, and C. F. Schmidt, “Twin optical traps for two-particle cross-correlation measurements: Eliminating cross-talk,” Rev. Sci. Instrum. |

15. | J.-C. Meiners and S. R. Quake, “Direct measurement of hydrodynamic cross correlations between two particles in an external potential,” Phys. Rev. Lett. |

16. | C. D. Saunter, G. D. Love, M. Johns, and J. Holmes, “FGPA technology for high-speed, low-cost adaptive optics,” vol. 6018 of |

17. | R. Di Leonardo, S. Keen, J. Leach, C. D. Saunter, G. D. Love, G. Ruocco, and M. J. Padgett, “Eigenmodes of a hydrodynamically coupled micron-size multiple-particle ring,” Phys. Rev. E |

**OCIS Codes**

(140.7010) Lasers and laser optics : Laser trapping

(170.4520) Medical optics and biotechnology : Optical confinement and manipulation

(350.4855) Other areas of optics : Optical tweezers or optical manipulation

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: June 27, 2008

Revised Manuscript: August 13, 2008

Manuscript Accepted: August 25, 2008

Published: September 2, 2008

**Virtual Issues**

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

**Citation**

Graham M. Gibson, Jonathan Leach, Stephen Keen, Amanda J. Wright, and Miles J. Padgett, "Measuring the accuracy of particle
position and force in optical tweezers
using high-speed video microscopy," Opt. Express **16**, 14561-14570 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-19-14561

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### References

- A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, "Observation of a single-beam gradient force optical trap for dielectric particles," Opt. Lett. 11, 288-290 (1986). [CrossRef] [PubMed]
- J. E. Molloy and M. J. Padgett, "Lights, action: optical tweezers," Contemp. Phys. 43, 241-258 (2002). [CrossRef]
- J.-C. Meiners and S. R. Quake, "Femtonewton force spectroscopy of single extended DNA molecules," Phys. Rev. Lett. 84, 5014-5017 (2000). [CrossRef] [PubMed]
- K. C. Neuman and S. M. Block, "Optical trapping," Rev. Sci. Instrum. 75, 2787-2809 (2004). [CrossRef]
- M. Polin, D. G. Grier, and S. R. Quake, "Anomalous vibrational dispersion in holographically trapped colloidal arrays," Phys. Rev. Lett. 96, 088101 (2006). [CrossRef] [PubMed]
- S. Keen, J. Leach, G. Gibson, and M. Padgett, "Comparison of a high-speed camera and a quadrant detector for measuring displacements in optical tweezers," J. Opt. A: Pure Appl. Opt. 9, S264-S266 (2007). [CrossRef]
- O. Otto, C. Gutsche, F. Kremer, and U. F. Keyser, "Optical tweezers with 2.5kHz bandwidth video detection for single-colloid electrophoresis," Rev. Sci. Instrum. 79, 023710 (2008). [CrossRef] [PubMed]
- D. W. Allan, "Statistics of atomic frequency standards," Proc. IEEE 54, 221-230 (1966). [CrossRef]
- J. Leach, K. Wulff, G. Sinclair, P. Jordan, J. Courtial, L. Thomson, G. Gibson, K. Karunwi, J. Cooper, Z. J. Laczik, and M. Padgett, "Interactive approach to optical tweezers control," Appl. Opt. 45, 897-903 (2006). [CrossRef] [PubMed]
- G. Gibson, D. M. Carberry, G. Whyte, J. Leach, J. Courtial, J. C. Jackson, D. Robert, M. Miles, and M. Padgett, "Holographic assembly workstation for optical manipulation," J. Opt. A: Pure Appl. Opt. 10, 044009 (2008). [CrossRef]
- K. Berg-Sørensen and H. Flyvbjerg, "Power spectrum analysis for optical tweezers," Rev. Sci. Instrum. 75, 594-612 (2004). [CrossRef]
- J. R. Moffitt, Y. R. Chemla, D. Izhaky, and C. Bustamante, "Differential detection of dual traps improves the spatial resolution of optical tweezers," PNAS 103, 9006-9011 (2006). [CrossRef] [PubMed]
- M. Klein, M. Andersson, O. Axner, and E. F¨allman, "Dual-trap technique for reduction of low-frequency noise in force measuring optical tweezers," Appl. Opt. 46, 405-412 (2007). [CrossRef] [PubMed]
- M. Atakhorrami, K. M. Addas, and C. F. Schmidt, "Twin optical traps for two-particle cross-correlation measurements: Eliminating cross-talk," Rev. Sci. Instrum. 79, 043103 (2008). [CrossRef] [PubMed]
- J.-C. Meiners and S. R. Quake, "Direct measurement of hydrodynamic cross correlations between two particles in an external potential," Phys. Rev. Lett. 82, 2211-2214 (1999). [CrossRef]
- C. D. Saunter, G. D. Love,M. Johns, and J. Holmes, "FGPA technology for high-speed, low-cost adaptive optics," vol. 6018 of 5th International Workshop on Adaptive Optics for Industry and Medicine (SPIE, 2005).
- R. Di Leonardo, S. Keen, J. Leach, C. D. Saunter, G. D. Love, G. Ruocco, and M. J. Padgett, "Eigenmodes of a hydrodynamically coupled micron-size multiple-particle ring," Phys. Rev. E 76, 061402 (2007). [CrossRef]

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