## Holographic deconvolution microscopy for high-resolution particle tracking |

Optics Express, Vol. 19, Issue 17, pp. 16410-16417 (2011)

http://dx.doi.org/10.1364/OE.19.016410

Acrobat PDF (1640 KB)

### Abstract

Rayleigh-Sommerfeld back-propagation can be used to reconstruct the three-dimensional light field responsible for the recorded intensity in an in-line hologram. Deconvolving the volumetric reconstruction with an optimal kernel derived from the Rayleigh-Sommerfeld propagator itself emphasizes the objects responsible for the scattering pattern while suppressing both the propagating light and also such artifacts as the twin image. Bright features in the deconvolved volume may be identified with such objects as colloidal spheres and nanorods. Tracking their thermally-driven Brownian motion through multiple holographic video images provides estimates of the tracking resolution, which approaches 1 nm in all three dimensions.

© 2011 OSA

1. J. Sheng, E. Malkiel, and J. Katz, “Digital holographic microscope for measuring three-dimensional particle distributions and motions,” Appl. Opt. **45**, 3893–3901 (2006). [CrossRef] [PubMed]

2. S.-H. Lee and D. G. Grier, “Holographic microscopy of holographically trapped three-dimensional structures,” Opt. Express **15**, 1505–1512 (2007). [CrossRef] [PubMed]

3. 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**, 18275–18282 (2007). [CrossRef] [PubMed]

4. F. C. Cheong, B. Sun, R. Dreyfus, J. Amato-Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express **17**, 13071–13079 (2009). [CrossRef] [PubMed]

3. 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**, 18275–18282 (2007). [CrossRef] [PubMed]

5. F. C. Cheong, B. J. Krishnatreya, and D. G. Grier, “Strategies for three-dimensional particle tracking with holographic video microscopy,” Opt. Express **18**, 13563–13573 (2010). [CrossRef] [PubMed]

3. 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**, 18275–18282 (2007). [CrossRef] [PubMed]

4. F. C. Cheong, B. Sun, R. Dreyfus, J. Amato-Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express **17**, 13071–13079 (2009). [CrossRef] [PubMed]

6. F. C. Cheong, K. Xiao, and D. G. Grier, “Characterization of individual milk fat globules with holographic video microscopy,” J. Dairy Sci. **92**, 95–99 (2009). [CrossRef]

7. K. Xiao and D. G. Grier, “Multidimensional optical fractionation with holographic verification,” Phys. Rev. Lett. **104**, 028302 (2010). [CrossRef] [PubMed]

8. F. C. Cheong, K. Xiao, D. J. Pine, and D. G. Grier, “Holographic characterization of individual colloidal spheres’ porosities,” Soft Matter (to be published), DOI:. [CrossRef]

4. F. C. Cheong, B. Sun, R. Dreyfus, J. Amato-Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express **17**, 13071–13079 (2009). [CrossRef] [PubMed]

6. F. C. Cheong, K. Xiao, and D. G. Grier, “Characterization of individual milk fat globules with holographic video microscopy,” J. Dairy Sci. **92**, 95–99 (2009). [CrossRef]

1. J. Sheng, E. Malkiel, and J. Katz, “Digital holographic microscope for measuring three-dimensional particle distributions and motions,” Appl. Opt. **45**, 3893–3901 (2006). [CrossRef] [PubMed]

2. S.-H. Lee and D. G. Grier, “Holographic microscopy of holographically trapped three-dimensional structures,” Opt. Express **15**, 1505–1512 (2007). [CrossRef] [PubMed]

9. B. Rappaz, P. Marquet, E. Cuche, Y. Emery, C. Depeursinge, and P. J. Magistretti, “Measurement of the integral refractive index and dynamic cell morphometry of living cells with digital holographic microscopy,” Opt. Express **13**, 9361–9373 (2005). [CrossRef] [PubMed]

1. J. Sheng, E. Malkiel, and J. Katz, “Digital holographic microscope for measuring three-dimensional particle distributions and motions,” Appl. Opt. **45**, 3893–3901 (2006). [CrossRef] [PubMed]

2. S.-H. Lee and D. G. Grier, “Holographic microscopy of holographically trapped three-dimensional structures,” Opt. Express **15**, 1505–1512 (2007). [CrossRef] [PubMed]

5. F. C. Cheong, B. J. Krishnatreya, and D. G. Grier, “Strategies for three-dimensional particle tracking with holographic video microscopy,” Opt. Express **18**, 13563–13573 (2010). [CrossRef] [PubMed]

5. F. C. Cheong, B. J. Krishnatreya, and D. G. Grier, “Strategies for three-dimensional particle tracking with holographic video microscopy,” Opt. Express **18**, 13563–13573 (2010). [CrossRef] [PubMed]

**18**, 13563–13573 (2010). [CrossRef] [PubMed]

11. F. C. Cheong and D. G. Grier, “Rotational and translational diffusion of copper oxide nanorods measured with holographic video microscopy,” Opt. Express **18**, 6555–6562 (2010). [CrossRef] [PubMed]

**18**, 13563–13573 (2010). [CrossRef] [PubMed]

12. T. Latychevskaia, F. Gehri, and H.-W. Fink, “Depth-resolved holographic reconstructions by three-dimensional deconvolution,” Opt. Express **21**, 22527–22544 (2010). [CrossRef]

13. Y. Cotte, M. F. Toy, N. Pavillon, and C. Depeursinge, “Microscopy image resolution improvement by deconvolution of complex fields,” Opt. Express **18**, 19462–19478 (2010). [CrossRef] [PubMed]

**15**, 1505–1512 (2007). [CrossRef] [PubMed]

^{2}. Holographic images are captured by a low-noise gray-scale video camera (NEC TI 324A-II) and are recorded as an uncompressed digital video stream at 30 frames per second with a total system magnification of 135 nm/pixel. The camera’s 0.5 ms exposure time is fast enough to avoid measurable effects of motion blurring [4

**17**, 13071–13079 (2009). [CrossRef] [PubMed]

14. T. Savin and P. S. Doyle, “Role of finite exposure time on measuring an elastic modulus using microrheology,” Phys. Rev. E **71**, 041106 (2005). [CrossRef]

16. L. Dixon, F. C. Cheong, and D. G. Grier, “Holographic particle-streak velocimetry,” Opt. Express **19**, 4393–4398 (2011). [CrossRef] [PubMed]

*ẑ*with an real-valued amplitude

*E*

_{0}(

**r**) that may depend on position

**r**= (

*x,y*) in the transverse plane, and uniform polarization

ɛ ^

_{0}. The scattered wave, propagates in three dimensions with complex amplitude

*E*(

_{S}**r**,

*z*) and spatially varying polarization

*(*ɛ ^

**r**,

*z*). Their superposition in the focal plane yields the interference pattern Deliberately placing the scatterer well above the focal plane ensures both that polarization rotations are small,

*E*(

_{R}**r**,

*z*) =

*E*(

_{S}**r**,

*z*)/

*E*

_{0}(

**r**). Dropping |

*E*(

_{R}**r**, 0)|

^{2}from the definition of

*b*(

**r**) simplifies the analysis that follows at the cost of ignoring interference due to multiple scatterers. It therefore limits the complexity of the samples to which this formalism may be applied. In practice, the background image,

*I*

_{0}(

**r**), can be obtained either by moving the sample out of the field of view, or by taking a running median filter of a time-evolving sample.

*z*as the convolution of the scattered amplitude in the focal plane with the Rayleigh-Sommerfeld propagator [10] where

*R*

^{2}=

*r*

^{2}+

*z*

^{2}. The sign convention for

*z*accounts for the object’s position upstream of the focal plane. Equation (6) may be rewritten with the Fourier convolution theorem as where is the in-plane Fourier transform of

*E*(

_{R}**r**,

*z*) and where is the Fourier transform of

*h*(

**r**, –

*z*) [10, 17

17. G. C. Sherman, “Application of the convolution theorem to Rayleigh’s integral formulas,” J. Opt. Soc. Am. **57**, 546–547 (1967). [CrossRef] [PubMed]

18. U. Schnars and W. P. O. Jüptner, “Digital recording and reconstruction of holograms,” Meas. Sci. Technol. **13**, R85–R101 (2002). [CrossRef]

*b*(

**r**) is From this, may be recognized as the superposition of the scattered field at height

*z*above the focal plane and a spurious field due to the object’s mirror image in the focal plane, which is known as the twin image. The twin image’s influence on the reconstructed field may be reduced by moving the sample away from the focal plane.

*I*(

_{R}**r**,

*z*) = |

*E*(

_{R}**r**,

*z*)|

^{2}, is an estimate for the scattered light’s intensity at height

*z*above the focal plane. This reconstruction differs from a numerically refocused image [1

**45**, 3893–3901 (2006). [CrossRef] [PubMed]

11. F. C. Cheong and D. G. Grier, “Rotational and translational diffusion of copper oxide nanorods measured with holographic video microscopy,” Opt. Express **18**, 6555–6562 (2010). [CrossRef] [PubMed]

19. T. Yu, C. H. Sow, A. Gantimahapatruni, F. C. Cheong, Y. W. Zhu, K. C. Chin, X. J. Xu, C. T. Lim, Z. X. Shen, J. T. L. Thong, and A. T. S. Wee, “Patterning and fusion of CuO nanorods with a focused laser beam,” Nanotechnology **16**, 1238–1244 (2005). [CrossRef]

12. T. Latychevskaia, F. Gehri, and H.-W. Fink, “Depth-resolved holographic reconstructions by three-dimensional deconvolution,” Opt. Express **21**, 22527–22544 (2010). [CrossRef]

13. Y. Cotte, M. F. Toy, N. Pavillon, and C. Depeursinge, “Microscopy image resolution improvement by deconvolution of complex fields,” Opt. Express **18**, 19462–19478 (2010). [CrossRef] [PubMed]

*z*above the focal plane into the three-dimensional intensity distribution

_{p}*K*(

**r**,

*z*–

*z*) = |

_{p}*h*(

**r**,

*z*–

*z*)|

_{p}^{2}. The caustic’s twin image, similarly, is projected into

*K*(

**r**,

*z*+

*z*). Deconvolving with

_{p}*K*(

**r**,

*z*) therefore should reduce the full three-dimensional scattering pattern reconstructed from the hologram to the set of focal caustics created by the sample. Twin images also will be reduced to focal caustics, but on the other side of the effective focal plane, which will not be reconstructed, and so will not be seen. Using the Rayleigh-Sommerfeld point-spread function in this way also eliminates the need for calibration measurements [12

12. T. Latychevskaia, F. Gehri, and H.-W. Fink, “Depth-resolved holographic reconstructions by three-dimensional deconvolution,” Opt. Express **21**, 22527–22544 (2010). [CrossRef]

13. Y. Cotte, M. F. Toy, N. Pavillon, and C. Depeursinge, “Microscopy image resolution improvement by deconvolution of complex fields,” Opt. Express **18**, 19462–19478 (2010). [CrossRef] [PubMed]

*I*(

_{R}**r**,

*z*) with

*K*(

**r**, –

*z*) is most easily implemented with the Fourier convolution theorem: where

*Ĩ*(

_{R}*ρ*) is the three-dimensional Fourier transform of

*I*(

_{R}**r**,

*z*),

*K̃*(

*ρ*) is the Fourier transform of

*K*(

**r**, –

*z*), and

*χ*is a small factor chosen for numerical stability. The deconvolved intensity,

*I*(

_{D}**r**,

*z*) is obtained as the inverse Fourier transform of

*Ĩ*(

_{D}*ρ*). Figures 1(c) and 1(d) (Media 1) were obtained in this way from the data in Figs. 1(a) and 1(b), respectively, and are rendered with the same color and transparency tables. Blurring from artifacts and out-of-focus images is strongly suppressed in the deconvolved volumes, leaving distinct and well-resolved bright features that we associate with the focal caustics created by the samples. For example, the sample in Fig. 1(c) now can be recognized to consist of 9 colloidal spheres arranged in a body-centered cubic lattice, while that in Fig. 1(d) is quite clearly an inclined rod. These results were obtained with

*χ*= 0.01, and comparably good contrast and resolution were obtained for 0.005 ≤

*χ*< 0.05.

21. J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. **179**, 298–310 (1996). [CrossRef]

**18**, 13563–13573 (2010). [CrossRef] [PubMed]

11. F. C. Cheong and D. G. Grier, “Rotational and translational diffusion of copper oxide nanorods measured with holographic video microscopy,” Opt. Express **18**, 6555–6562 (2010). [CrossRef] [PubMed]

**21**, 22527–22544 (2010). [CrossRef]

*a priori*knowledge.

**18**, 13563–13573 (2010). [CrossRef] [PubMed]

**18**, 6555–6562 (2010). [CrossRef] [PubMed]

14. T. Savin and P. S. Doyle, “Role of finite exposure time on measuring an elastic modulus using microrheology,” Phys. Rev. E **71**, 041106 (2005). [CrossRef]

15. T. Savin and P. S. Doyle, “Static and dynamic errors in particle tracking microrheology,” Biophys. J. **88**, 623–638 (2005). [CrossRef]

21. J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. **179**, 298–310 (1996). [CrossRef]

*r̂*, increases linearly with time according to the familiar Einstein-Smoluchowski result, but is off-set by the measurement error

_{j}*ɛ*for centroid location along that axis [21

_{j}21. J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. **179**, 298–310 (1996). [CrossRef]

14. T. Savin and P. S. Doyle, “Role of finite exposure time on measuring an elastic modulus using microrheology,” Phys. Rev. E **71**, 041106 (2005). [CrossRef]

16. L. Dixon, F. C. Cheong, and D. G. Grier, “Holographic particle-streak velocimetry,” Opt. Express **19**, 4393–4398 (2011). [CrossRef] [PubMed]

*μ*m diameter colloidal polystyrene sphere (Bangs Laboratories, lot number 912) diffusing in water at room temperature,

*T*= 296 K. The three single-axis diffusion coefficients,

*D*= 0.615 ± 0.157

_{j}*μ*m

^{2}/

*s*, (

*j*=

*x,y,z*) are consistent with each other and also are consistent with the Stokes-Einstein result

*D*=

*k*/(6

_{B}T*πηa*) given

_{p}*η*= 0.89 cP and using the radius

*a*= 0.428 ± 0.001

_{p}*μ*m obtained from Lorenz-Mie characterization [3

**15**, 18275–18282 (2007). [CrossRef] [PubMed]

*ɛ*= 6 ± 1 nm,

_{x}*ɛ*= 7 ± 1 nm and

_{y}*ɛ*= 10 ± 1 nm, are a factor of two smaller than the corresponding errors obtained with Rayleigh-Sommerfeld reconstruction without deconvolution [5

_{z}**18**, 13563–13573 (2010). [CrossRef] [PubMed]

**18**, 13563–13573 (2010). [CrossRef] [PubMed]

**18**, 13563–13573 (2010). [CrossRef] [PubMed]

**18**, 13563–13573 (2010). [CrossRef] [PubMed]

*L*= 5.0 ± 0.3

*μ*m, whose length remains unchanged through a sequence of 5,000 snapshots as the nanorod rotates in three-dimensions. It is tempting to identify the center of this cylinder with the nanorod’s position, and the orientation of its axis with the nanorod’s orientation. Figure 2(b) shows the mean-squared displacement of the orientational unit vector

*ŝ*(

*t*) obtained in this way. The nanorod’s apparent orientation and center-of-mass position

**r**(

*t*) are extracted from the deconvolved reconstruction using a peak-tracking skeletonization algorithm originally developed [11

**18**, 6555–6562 (2010). [CrossRef] [PubMed]

*D*= 0.094 ± 0.009 s

_{r}^{−1}and the measurement error

*ɛ*= sin Δ

_{s}*θ*= 0.036 ± 0.020 in the nanorod’s three-dimensional orientation. The latter figure is consistent with an error of Δ

*θ*= 1° in the rod’s orientation, which improves on previous results by more than a factor of two.

22. Y. Han, A. M. Alsayed, M. Nobili, J. Zhang, T. C. Lubensky, and A. G. Yodh, “Brownian motion of an ellipsoid,” Science **314**, 626–630 (2009). [CrossRef]

*D*

_{||}= 0.411 ± 0.003

*μ*m

^{2}/s and

*D*

_{⊥}= 0.241 ± 0.012

*μ*m

^{2}/s.

*L*and diameter

*σ*, and also on a geometric factor

*γ*[23]. From these, we obtain

*L*= 5.12 ± 0.025

*μ*m, which is consistent with the optical measurement,

*σ*= 241 ± 12 nm, and

*γ*= 0.25. The associated tracking error for the rod’s centroid is 10 nm in the plane and 50 nm along the optical axis, averaged over orientations.

**18**, 13563–13573 (2010). [CrossRef] [PubMed]

**18**, 13563–13573 (2010). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | J. Sheng, E. Malkiel, and J. Katz, “Digital holographic microscope for measuring three-dimensional particle distributions and motions,” Appl. Opt. |

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

3. | 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 |

4. | F. C. Cheong, B. Sun, R. Dreyfus, J. Amato-Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express |

5. | F. C. Cheong, B. J. Krishnatreya, and D. G. Grier, “Strategies for three-dimensional particle tracking with holographic video microscopy,” Opt. Express |

6. | F. C. Cheong, K. Xiao, and D. G. Grier, “Characterization of individual milk fat globules with holographic video microscopy,” J. Dairy Sci. |

7. | K. Xiao and D. G. Grier, “Multidimensional optical fractionation with holographic verification,” Phys. Rev. Lett. |

8. | F. C. Cheong, K. Xiao, D. J. Pine, and D. G. Grier, “Holographic characterization of individual colloidal spheres’ porosities,” Soft Matter (to be published), DOI:. [CrossRef] |

9. | B. Rappaz, P. Marquet, E. Cuche, Y. Emery, C. Depeursinge, and P. J. Magistretti, “Measurement of the integral refractive index and dynamic cell morphometry of living cells with digital holographic microscopy,” Opt. Express |

10. | J. W. Goodman, |

11. | F. C. Cheong and D. G. Grier, “Rotational and translational diffusion of copper oxide nanorods measured with holographic video microscopy,” Opt. Express |

12. | T. Latychevskaia, F. Gehri, and H.-W. Fink, “Depth-resolved holographic reconstructions by three-dimensional deconvolution,” Opt. Express |

13. | Y. Cotte, M. F. Toy, N. Pavillon, and C. Depeursinge, “Microscopy image resolution improvement by deconvolution of complex fields,” Opt. Express |

14. | T. Savin and P. S. Doyle, “Role of finite exposure time on measuring an elastic modulus using microrheology,” Phys. Rev. E |

15. | T. Savin and P. S. Doyle, “Static and dynamic errors in particle tracking microrheology,” Biophys. J. |

16. | L. Dixon, F. C. Cheong, and D. G. Grier, “Holographic particle-streak velocimetry,” Opt. Express |

17. | G. C. Sherman, “Application of the convolution theorem to Rayleigh’s integral formulas,” J. Opt. Soc. Am. |

18. | U. Schnars and W. P. O. Jüptner, “Digital recording and reconstruction of holograms,” Meas. Sci. Technol. |

19. | T. Yu, C. H. Sow, A. Gantimahapatruni, F. C. Cheong, Y. W. Zhu, K. C. Chin, X. J. Xu, C. T. Lim, Z. X. Shen, J. T. L. Thong, and A. T. S. Wee, “Patterning and fusion of CuO nanorods with a focused laser beam,” Nanotechnology |

20. | J. F. Nye, |

21. | J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. |

22. | Y. Han, A. M. Alsayed, M. Nobili, J. Zhang, T. C. Lubensky, and A. G. Yodh, “Brownian motion of an ellipsoid,” Science |

23. | M. Doi and S. F. Edwards, |

24. | C. J. Hernandez and T. G. Mason, “Colloidal alphabet soup: Monodisperse dispersions of shape-designed LithoParticles,” J. Phys. Chem. |

**OCIS Codes**

(100.2960) Image processing : Image analysis

(180.6900) Microscopy : Three-dimensional microscopy

(350.4990) Other areas of optics : Particles

(090.1995) Holography : Digital holography

**ToC Category:**

Holography

**History**

Original Manuscript: May 16, 2011

Revised Manuscript: July 19, 2011

Manuscript Accepted: August 1, 2011

Published: August 11, 2011

**Virtual Issues**

Vol. 6, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Lisa Dixon, Fook Chiong Cheong, and David G. Grier, "Holographic deconvolution microscopy for high-resolution particle tracking," Opt. Express **19**, 16410-16417 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-17-16410

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### References

- J. Sheng, E. Malkiel, and J. Katz, “Digital holographic microscope for measuring three-dimensional particle distributions and motions,” Appl. Opt. 45, 3893–3901 (2006). [CrossRef] [PubMed]
- S.-H. Lee and D. G. Grier, “Holographic microscopy of holographically trapped three-dimensional structures,” Opt. Express 15, 1505–1512 (2007). [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, 18275–18282 (2007). [CrossRef] [PubMed]
- F. C. Cheong, B. Sun, R. Dreyfus, J. Amato-Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express 17, 13071–13079 (2009). [CrossRef] [PubMed]
- F. C. Cheong, B. J. Krishnatreya, and D. G. Grier, “Strategies for three-dimensional particle tracking with holographic video microscopy,” Opt. Express 18, 13563–13573 (2010). [CrossRef] [PubMed]
- F. C. Cheong, K. Xiao, and D. G. Grier, “Characterization of individual milk fat globules with holographic video microscopy,” J. Dairy Sci. 92, 95–99 (2009). [CrossRef]
- K. Xiao and D. G. Grier, “Multidimensional optical fractionation with holographic verification,” Phys. Rev. Lett. 104, 028302 (2010). [CrossRef] [PubMed]
- F. C. Cheong, K. Xiao, D. J. Pine, and D. G. Grier, “Holographic characterization of individual colloidal spheres’ porosities,” Soft Matter (to be published), DOI: . [CrossRef]
- B. Rappaz, P. Marquet, E. Cuche, Y. Emery, C. Depeursinge, and P. J. Magistretti, “Measurement of the integral refractive index and dynamic cell morphometry of living cells with digital holographic microscopy,” Opt. Express 13, 9361–9373 (2005). [CrossRef] [PubMed]
- J. W. Goodman, Introduction to Fourier Optics , 3rd ed. (McGraw-Hill, 2005).
- F. C. Cheong and D. G. Grier, “Rotational and translational diffusion of copper oxide nanorods measured with holographic video microscopy,” Opt. Express 18, 6555–6562 (2010). [CrossRef] [PubMed]
- T. Latychevskaia, F. Gehri, and H.-W. Fink, “Depth-resolved holographic reconstructions by three-dimensional deconvolution,” Opt. Express 21, 22527–22544 (2010). [CrossRef]
- Y. Cotte, M. F. Toy, N. Pavillon, and C. Depeursinge, “Microscopy image resolution improvement by deconvolution of complex fields,” Opt. Express 18, 19462–19478 (2010). [CrossRef] [PubMed]
- T. Savin and P. S. Doyle, “Role of finite exposure time on measuring an elastic modulus using microrheology,” Phys. Rev. E 71, 041106 (2005). [CrossRef]
- T. Savin and P. S. Doyle, “Static and dynamic errors in particle tracking microrheology,” Biophys. J. 88, 623–638 (2005). [CrossRef]
- L. Dixon, F. C. Cheong, and D. G. Grier, “Holographic particle-streak velocimetry,” Opt. Express 19, 4393–4398 (2011). [CrossRef] [PubMed]
- G. C. Sherman, “Application of the convolution theorem to Rayleigh’s integral formulas,” J. Opt. Soc. Am. 57, 546–547 (1967). [CrossRef] [PubMed]
- U. Schnars and W. P. O. Jüptner, “Digital recording and reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002). [CrossRef]
- T. Yu, C. H. Sow, A. Gantimahapatruni, F. C. Cheong, Y. W. Zhu, K. C. Chin, X. J. Xu, C. T. Lim, Z. X. Shen, J. T. L. Thong, and A. T. S. Wee, “Patterning and fusion of CuO nanorods with a focused laser beam,” Nanotechnology 16, 1238–1244 (2005). [CrossRef]
- J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics Publishing, 1999).
- J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179, 298–310 (1996). [CrossRef]
- Y. Han, A. M. Alsayed, M. Nobili, J. Zhang, T. C. Lubensky, and A. G. Yodh, “Brownian motion of an ellipsoid,” Science 314, 626–630 (2009). [CrossRef]
- M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Clarendon Press, 1986).
- C. J. Hernandez and T. G. Mason, “Colloidal alphabet soup: Monodisperse dispersions of shape-designed LithoParticles,” J. Phys. Chem. 111, 4477–4480 (2007).

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