## Three dimensional tracking of fluorescent microparticles using a photon-limited double-helix response system

Optics Express, Vol. 16, Issue 26, pp. 22048-22057 (2008)

http://dx.doi.org/10.1364/OE.16.022048

Acrobat PDF (1053 KB)

### Abstract

We demonstrate three-dimensional tracking of fluorescent microparticles, with a computational optical system whose point spread function (PSF) has been engineered to have two twisting lobes along the optical axis, generating a three-dimensional (3D) double-helix (DH) PSF. An information theoretical comparison in photon limited systems shows that the DH-PSF delivers higher Fisher information for 3D localization than the standard PSF. Hence, DH-PSF systems provide better position estimation accuracy. Experiments demonstrate average position estimation accuracies under 14nm and 37nm in the transverse and axial dimensions respectively. The system determines the 3D position of multiple particles with a single image and tracks them over time while providing their velocities.

© 2008 Optical Society of America

## 1. Introduction

1. J. A. Steyer and W. Almers, “Tracking single secretory granules in live chromaffin cells by evanescentfield fluorescence microscopy,” Biophys. J. **76**, 2262–2271 (1999). [CrossRef] [PubMed]

2. D. Li, J. Xiong, A. Qu, and T. Xu, “Three-dimensional tracking of single secretory granules in live pc12 cells,” Biophys. J. **87**, 1991–2001 (2004). [CrossRef] [PubMed]

3. R. M. Dickson, A. B. Cubitt, R. Y. Tsien, and W. E. Moerner, “On/off blinking and switching behaviour of single molecules of green fluorescent protein,” Nature **388**, 355 (1997) [CrossRef] [PubMed]

4. G. J. Schütz, M. Axmann, and H. Schindler, “Imaging single molecules in three dimensions,” Sing. Mol. **2**, 69–74 (2001). [CrossRef]

5. J. A. Steinkamp, J. S. Wilson, G. C. Saunders, and C. C. Stewart, “Phagocytosis: flow cytometric quantitation with fluorescent microspheres,” Science **215**, 64–66 (1982). [CrossRef] [PubMed]

6. R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. **82**, 2775–2783 (2002). [CrossRef] [PubMed]

7. M. Speidel, A. Jonas, and E.-L. Florin, “Three-dimensional tracking of fluorescent nanoparticles with subnanometer precision by use of off-focus imaging,” Opt. Lett. **28**, 69 (2003). [CrossRef] [PubMed]

8. H. P. Kao and A. S. Verkman, “Tracking of single fluorescent particles in three dimensions: use of cylindrical optics to encode particle position,” Biophys. J. **67**, 1291–1300 (1994). [CrossRef] [PubMed]

10. B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-Dimensional Super-Resolution Imaging by Stochastic Optical Reconstruction Microscopy,” Science **319**, 5864, 810–813 (2008). [CrossRef] [PubMed]

11. S. -H. Lee, Y. Roichman, G. -R. Yi, S. -H. Kim, S. -M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express **15**, 18275 (2007). [CrossRef] [PubMed]

12. S. -H. Lee and D. G. Grier, “Holographic microscopy of holographically trapped three-dimensional structures,” Opt. Express **15**, 1505 (2007). [CrossRef] [PubMed]

13. J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nature Photon. **2**, 190–195 (2008). [CrossRef]

*photon-unlimited*monochromatic wide-field microscope presenting a double-helix (DH) PSF, which exhibits two lobes that rotate continuously with defocus. An information theoretical analysis showed that the DH-PSF is fundamentally better suited than the standard diffraction-limited PSF for 3D position localization in photon-unlimited systems with either Gaussian or Poisson noise.

*photon-limited*DH-PSF systems. In order to compare the 3D position localization accuracies of systems with different PSFs, we introduce a new quality measure based on the average Fisher information over the 3D volume of interest. Using this measure, we show that in the photon-limited case, DH-PSF systems carry higher average Fisher information than the standard PSF systems. We then demonstrate simultaneous 3D position estimations of multiple fluorescent particles using a single image with accuracies in the nanometer regime. By acquiring multiple images at periodic intervals, we track moving fluorescent particles in three dimensions, and calculate their velocities.

### 1.1. Double-helix point spread function (DH-PSF)

15. S. R. P. Pavani and R. Piestun, “High-efficiency rotating point spread functions,” Opt. Express **16**, 3484–3489 (2008). [CrossRef] [PubMed]

18. R. Piestun, Y. Y. Schechner, and J. Shamir, “Propagation-invariant wave fields with finite energy,” J. Opt. Soc. Am. A **17**, 294–303 (2000). [CrossRef]

17. Y. Y. Schechner, R. Piestun, and J. Shamir, “Wave propagation with rotating intensity distributions,” Phys. Rev. E **54**, R50–R53 (1996). [CrossRef]

16. A. Greengard, Y. Y. Schechner, and R. Piestun, “Depth from diffracted rotation,” Opt. Lett. **31**, 181–183 (2006). [CrossRef] [PubMed]

15. S. R. P. Pavani and R. Piestun, “High-efficiency rotating point spread functions,” Opt. Express **16**, 3484–3489 (2008). [CrossRef] [PubMed]

## 2. Cramer-Rao bounds in photon limited systems

16. A. Greengard, Y. Y. Schechner, and R. Piestun, “Depth from diffracted rotation,” Opt. Lett. **31**, 181–183 (2006). [CrossRef] [PubMed]

21. R. J. Ober, S. Ram, and S.E. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. **86**, 1185–1200 (2004). [CrossRef] [PubMed]

*[14, 19], which is calculated as follows:*

**I***θ*=[X, Y, Z], the indices

*m*and

*n*are either 1, 2, or 3,

*E*is the expectation, and

*p*(

_{i,j}*k*|

*θ*) is the probability density function for the pixel in

*i*

^{th}row and

*j*

^{th}column.

*photon-unlimited*systems, such as bright-field microscopes, where increasing the illumination intensity increases the intensity of the detected PSF.

*photon-limited*systems, the detected PSF intensity cannot be arbitrarily increased. Here we call photon-limited systems those in which the illumination intensity cannot be arbitrarily increased and those in which even an increase in illumination intensity will not increase the detected intensity.

*CRB*

_{X},

*CRB*, and

_{Y}*CRB*, as functions of axial distance.

_{Z}*CRB*=

_{3D}*CRB*+

_{X}*CRB*+

_{Y}*CRB*, and its average over the axial region of interest,

_{Z}*CRB*largely mimics

_{3D}*CRB*, because for a 0.45NA objective,

_{Z}*CRB*is much higher than

_{Z}*CRB*and

_{X}*CRB*. The DH-PSF performs better in the regions A and C of the plot, while the standard PSF does better in region B. CRBAVG of the standard PSF is infinity, and that of the DH-PSF is 239 nm

_{Y}^{2}, which corresponds to an average combined standard deviation of 15.5nm in all three dimensions for a 0.45NA objective. DH-PSF is consequently better suited than the standard PSF for photon limited systems with Poisson noise. Moreover, the

*CRB*for the DH-PSF is more uniform throughout the whole range implying that a high 3D accuracy can be achieved over a long axial range.

_{3D}### 2.1. Practical DH-PSF estimator

## 3. Experiment

^{th}order response of the SLM by pushing the DH-PSF image to the SLM’s 1

^{st}order. The undiffracted 0

^{th}order was essentially the same as a standard PSF image as it was unaffected by the DH-PSF mask.

*The single-image*standard deviations were (

*σ*)=(14nm,13nm,37nm), on an average among the studied particles (see appendix for details). Because these microspheres were fixed, we could average the 100 estimations. The standard deviation of the average estimate was (

_{X},σ_{Y},σ_{Z}*σ*)=(3nm, 3nm, 6nm) on an average for the four particles.

_{x}̄,σ_{Y}̄,σ_{Z}̄*σ*)=(0.5nm,0.9nm,0.7nm) for an in-focus particle. These accuracies can potentially be achieved using more complex estimators, carefully calibrated detectors, and a more stable optical setup

_{X},σ_{Y},σ_{Z}^{1}.

### 3.1 Tracking experiment

## 6. Conclusion

## Appendix

## Calibration and detailed estimations

*X*̄,

*Y*̄,

*Z*̄) and standard deviations for all four particles in Fig. 3.

*σ*, and

_{X},σ_{Y}*σ*refer to the standard deviations along the X, Y, and Z dimensions, respectively.

_{Z}*σ*and

_{X}*σ*of a particle were computed from the standard deviation (

_{Y}*σ*) of the midpoint (

_{Xm},σ_{Ym}*X*) of the lobes of the particle, and the standard deviation (

_{m},Y_{m}*σ*is computed from the standard deviation (

_{Z}*σ*) of the rotation angle (

_{Za}*Z*) estimation, and the standard deviation (

_{a}*Zcl*) averaged over the 20 measurements. Specifically,

*σ*, and

_{Xm},σ_{Ym}*σ*are obtained as the standard deviation of the 100 estimates.

_{Za}*N*estimates is smaller than the standard deviation of each of

*N*estimates by a factor of (

*N*)1/2. Because the correction and calibration plots are obtained from an average of 20 estimates, the standard deviations (

^{1/2}. However,

*σ*, and

_{Xm}, σ_{Ym}*σ*are the standard deviations of a single estimate.

_{Za}*σ*,

_{X}*σ*, and

_{Y}*σ*listed in the table therefore represent single-image position estimation standard deviations. In other words,

_{Z}*σ*, and

_{X}, σ_{Y}*σ*are the accuracies when only one image is used for position estimation.

_{Z}*Xm,Ym*, and

*Za*from 100 images are averaged, the average estimates (

^{1/2}reduction in standard deviation (

*X*̄,

*Y*̄,

*Z*̄) are

_{X}̄,σ

_{Y}̄, and σ

_{Z}̄ for all four particles were 3nm, 3nm, and 6nm, respectively.

## Footnotes

1 | In addition to the photon noise considered in the CRB calculations, the above experimental standard deviations are also affected by the nanoscale vibrations in the optical setup caused by components such as the water-cooled Argon laser and the rotating diffuser. Tighter vibration control conditions could help approach the DH-PSF CRB limit for a given SNR. |

## Acknowledgments

## References and links

1. | J. A. Steyer and W. Almers, “Tracking single secretory granules in live chromaffin cells by evanescentfield fluorescence microscopy,” Biophys. J. |

2. | D. Li, J. Xiong, A. Qu, and T. Xu, “Three-dimensional tracking of single secretory granules in live pc12 cells,” Biophys. J. |

3. | R. M. Dickson, A. B. Cubitt, R. Y. Tsien, and W. E. Moerner, “On/off blinking and switching behaviour of single molecules of green fluorescent protein,” Nature |

4. | G. J. Schütz, M. Axmann, and H. Schindler, “Imaging single molecules in three dimensions,” Sing. Mol. |

5. | J. A. Steinkamp, J. S. Wilson, G. C. Saunders, and C. C. Stewart, “Phagocytosis: flow cytometric quantitation with fluorescent microspheres,” Science |

6. | R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. |

7. | M. Speidel, A. Jonas, and E.-L. Florin, “Three-dimensional tracking of fluorescent nanoparticles with subnanometer precision by use of off-focus imaging,” Opt. Lett. |

8. | H. P. Kao and A. S. Verkman, “Tracking of single fluorescent particles in three dimensions: use of cylindrical optics to encode particle position,” Biophys. J. |

9. | L. Holtzer, T. Meckel, and T. Schmidt, “Nanometric three-dimensional tracking of individual quantum dots in cells,” Appl. Phys. Lett. |

10. | B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-Dimensional Super-Resolution Imaging by Stochastic Optical Reconstruction Microscopy,” Science |

11. | S. -H. Lee, Y. Roichman, G. -R. Yi, S. -H. Kim, S. -M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express |

12. | S. -H. Lee and D. G. Grier, “Holographic microscopy of holographically trapped three-dimensional structures,” Opt. Express |

13. | J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nature Photon. |

14. | S. R. P. Pavani, A. Greengard, and R. Piestun, “Three-dimensional localization with nanometer accuracy using a double-helix point spread function system,” (submitted). |

15. | S. R. P. Pavani and R. Piestun, “High-efficiency rotating point spread functions,” Opt. Express |

16. | A. Greengard, Y. Y. Schechner, and R. Piestun, “Depth from diffracted rotation,” Opt. Lett. |

17. | Y. Y. Schechner, R. Piestun, and J. Shamir, “Wave propagation with rotating intensity distributions,” Phys. Rev. E |

18. | R. Piestun, Y. Y. Schechner, and J. Shamir, “Propagation-invariant wave fields with finite energy,” J. Opt. Soc. Am. A |

19. | S. M. Kay, |

20. | T. M. Cover and J. A. Thomas, |

21. | R. J. Ober, S. Ram, and S.E. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. |

**OCIS Codes**

(110.4850) Imaging systems : Optical transfer functions

(110.6880) Imaging systems : Three-dimensional image acquisition

(150.5670) Machine vision : Range finding

(180.2520) Microscopy : Fluorescence microscopy

(110.1758) Imaging systems : Computational imaging

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: October 8, 2008

Revised Manuscript: December 16, 2008

Manuscript Accepted: December 17, 2008

Published: December 19, 2008

**Virtual Issues**

Vol. 4, Iss. 2 *Virtual Journal for Biomedical Optics*

**Citation**

Sri Rama Prasanna Pavani and Rafael Piestun, "Three dimensional tracking of fluorescent microparticles using a photon-limited double-helix response system," Opt. Express **16**, 22048-22057 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-26-22048

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### References

- J. A. Steyer and W. Almers, "Tracking single secretory granules in live chromaffin cells by evanescent-field fluorescence microscopy," Biophys. J. 76, 2262-2271 (1999). [CrossRef] [PubMed]
- D. Li, J. Xiong, A. Qu, and T. Xu, "Three-dimensional tracking of single secretory granules in live pc12 cells," Biophys. J. 87, 1991-2001 (2004). [CrossRef] [PubMed]
- R. M. Dickson, A. B. Cubitt, R. Y. Tsien, and W. E. Moerner, "On/off blinking and switching behaviour of single molecules of green fluorescent protein," Nature 388, 355 (1997) [CrossRef] [PubMed]
- G. J. Schütz, M. Axmann, and H. Schindler, "Imaging single molecules in three dimensions," Sing. Mol. 2,69-74 (2001). [CrossRef]
- J. A. Steinkamp, J. S. Wilson, G. C. Saunders, and C. C. Stewart, "Phagocytosis: flow cytometric quantitation with fluorescent microspheres," Science 215, 64-66 (1982). [CrossRef] [PubMed]
- R. E. Thompson, D. R. Larson, and W. W. Webb, "Precise nanometer localization analysis for individual fluorescent probes," Biophys. J. 82,2775-2783 (2002). [CrossRef] [PubMed]
- M. Speidel, A. Jonas, and E.-L. Florin, "Three-dimensional tracking of fluorescent nanoparticles with subnanometer precision by use of off-focus imaging," Opt. Lett. 28, 69 (2003). [CrossRef] [PubMed]
- H. P. Kao and A. S. Verkman, "Tracking of single fluorescent particles in three dimensions: use of cylindrical optics to encode particle position," Biophys. J. 67, 1291-1300 (1994). [CrossRef] [PubMed]
- L. Holtzer, T. Meckel, and T. Schmidt, "Nanometric three-dimensional tracking of individual quantum dots in cells," Appl. Phys. Lett. 90, 053902 (2007). [CrossRef]
- B. Huang, W. Wang, M. Bates, and X. Zhuang, "Three-Dimensional Super-Resolution Imaging by Stochastic Optical Reconstruction Microscopy," Science 319, 5864, 810-813 (2008). [CrossRef] [PubMed]
- S. -H. Lee, Y. Roichman, G. -R. Yi, S. -H. Kim, S. -M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, "Characterizing and tracking single colloidal particles with video holographic microscopy," Opt. Express 15, 18275 (2007). [CrossRef] [PubMed]
- S. -H. Lee and D. G. Grier, "Holographic microscopy of holographically trapped three-dimensional structures," Opt. Express 15, 1505 (2007). [CrossRef] [PubMed]
- J. Rosen and G. Brooker, "Non-scanning motionless fluorescence three-dimensional holographic microscopy," Nature Photon. 2, 190-195 (2008). [CrossRef]
- S. R. P. Pavani, A. Greengard, and R. Piestun, "Three-dimensional localization with nanometer accuracy using a double-helix point spread function system," (submitted).
- S. R. P. Pavani and R. Piestun, "High-efficiency rotating point spread functions," Opt. Express 16, 3484-3489 (2008). [CrossRef] [PubMed]
- A. Greengard, Y. Y. Schechner, and R. Piestun, "Depth from diffracted rotation," Opt. Lett. 31, 181-183 (2006). [CrossRef] [PubMed]
- Y. Y. Schechner, R. Piestun, and J. Shamir, "Wave propagation with rotating intensity distributions," Phys. Rev. E 54, R50-R53 (1996). [CrossRef]
- R. Piestun, Y. Y. Schechner, and J. Shamir, "Propagation-invariant wave fields with finite energy," J. Opt. Soc. Am. A 17, 294-303 (2000). [CrossRef]
- S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 1993).
- T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley-Interscience, 1991). [CrossRef]
- R. J. Ober, S. Ram, and S.E. Ward, "Localization accuracy in single-molecule microscopy," Biophys. J. 86, 1185-1200 (2004). [CrossRef] [PubMed]

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