## Strategies for three-dimensional particle tracking with holographic video microscopy

Optics Express, Vol. 18, Issue 13, pp. 13563-13573 (2010)

http://dx.doi.org/10.1364/OE.18.013563

Acrobat PDF (1350 KB)

### Abstract

The video stream captured by an in-line holographic microscope can be analyzed on a frame-by-frame basis to track individual colloidal particles’ three-dimensional motions with nanometer resolution. In this work, we compare the performance of two complementary analysis techniques, one based on fitting to the exact Lorenz-Mie theory and the other based on phenomenological interpretation of the scattered light field reconstructed with Rayleigh-Sommerfeld back-propagation. Although Lorenz-Mie tracking provides more information and is inherently more precise, Rayleigh-Sommerfeld reconstruction is faster and more general. The two techniques agree quantitatively on colloidal spheres’ in-plane positions. Their systematic differences in axial tracking can be explained in terms of the illuminated objects’ light scattering properties.

© 2010 Optical Society of America

## 1. Introduction

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

2. J. C. Crocker and D. G. Grier, “Microscopic measurement of the pair interaction potential of charge-stabilized colloid,” Phys. Rev. Lett. **73**, 352–355 (1994). [CrossRef] [PubMed]

3. G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans, “Experimental demonstration of violations of the second law of thermodynmaics for small systems and short time scales,” Phys. Rev. Lett. **89**, 050601 (2002). [CrossRef] [PubMed]

4. T. G. Mason, K. Ganesan, J. H. van Zanten, D. Wirtz, and S. C. Kuo, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. **79**, 3282–3285 (1997). [CrossRef]

5. T. T. Perkins, D. E. Smith, R. G. Larson, and S. Chu, “Stretching of a single tethered polymer in a uniform flow,” Science **268**, 83–87 (1995). [CrossRef] [PubMed]

6. C. Gosse and V. Croquette, “Magnetic tweezers: Micromanipulation and force measurement at the molecular level,” Biophys. J. **82**, 3314–3329 (2002). [CrossRef] [PubMed]

7. 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]

10. F. C. Cheong, B. Sun, R. Dreyfus, A. Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express **17**, 13,071–13,079 (2009). [CrossRef]

7. 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]

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

13. J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. **45**, 836–850 (2006). [CrossRef] [PubMed]

14. 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]

7. 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]

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

14. 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]

## 2. In-line Holographic Video Microscopy

*λ*= 632.8 nm) replaces the conventional incandescent illuminator. The resulting irradiance of roughly 10 nW/

*µ*m

^{2}is comparable to that of conventional microscope illumination and is too weak to exert measurable forces on the illuminated objects or to heat the sample appreciably. An object at position

**r**

_{p}scatters a small portion of the incident plane wave, as depicted in Fig. 1. The scattered light then propagates to the focal plane of the microscope, where it interferes with the unscattered portion of the laser beam. The resulting interference pattern is magnified by the microscope’s objective lens (Nikon Plan Apo, 100×, numerical aperture 1.4, oil immersion) and projected by a video eyepiece (0.63×) onto a CCD camera (NEC TI-324AII), which records its intensity at 30 frames/s with an exposure time of 1 ms. The resulting video signal is recorded as uncompressed digital video with a digital video recorder (Pioneer DVR-520H) before being analyzed.

*µ*m/pixel. When used with conventional illumination and analyzed with standard methods of digital video microscopy [1

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

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

*µ*m by the limited depth of focus of the high-numerical-aperture objective lens. Illuminating the sample instead with coherent light provides access to more information at much higher resolution and over a much larger axial range, without requiring detailed calibrations. Extracting this information, however, requires a substantially more sophisticated analysis strategy.

15. F. Dubois, L. Joannes, and J. C. Legros, “Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence,” Appl. Opt. **38**, 7085–7094 (1999). [CrossRef]

16. J. Garcia-Sucerquia, J. H. Ramírez, and R. Castaneda, “Incoherent recovering of the spatial resolution in digital holography,” Opt. Commun. **260**, 62–67 (2006). [CrossRef]

## 3. Holographic Particle Tracking

*ẑ*with wavenumber

*k*= 2

*πn*/

_{m}*λ*in a medium of refractive index

*n*. Both the transverse amplitude profile

_{m}*u*

_{0}(

**r**) and the polarization

ε ^

_{0}are assumed to be independent of

*z*. The wave scattered by the sample,

*E*(

_{s}**r**,

*z*) and spatially varying polarization

*(*ε ^

**r**,

*z*). The measured intensity at point

**r**in the focal plane is due to the superposition of the incident and scattered waves,

*I*(

**r**) with a measured background image [8

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

10. F. C. Cheong, B. Sun, R. Dreyfus, A. Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express **17**, 13,071–13,079 (2009). [CrossRef]

11. 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]

14. 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]

ε ^

^{*}

_{0}·

*(*ε ^

**r**,0) ≈ 1. This approximation can be improved by deliberately defocusing the microscope. An optically isotropic sample’s normalized hologram then is described by

*E*(

_{R}**r**,

*z*) =

*E*(

_{s}**r**,

*z*)/

*u*

_{0}(

**r**).

### 3.1. Lorenz-Mie Fitting

**r**in the focal plane due to an object at

**r**

_{p}relative to the center of the focal plane may be written as

**f**

_{s}(

*k*

**r**), a normalized hologram may be fit to Eq. (7) for the position of the particle

**r**

_{p}as well as any free parameters in the scattering function [9].

**N**

^{(3)}

_{e1n}(

*k*

**r**) and

**M**

^{(3)}

_{o1n}(

*k*

**r**), whose coefficients,

*α*and

_{n}*β*, depend on the size, shape, composition, and orientation of the object and on the structure of the illuminating field. For a homogeneous isotropic sphere of radius

_{n}*a*and refractive index

_{p}*n*illuminated by a linearly polarized plane wave, the expansion coefficients are expressed in terms of spherical Bessel functions and spherical Hankel functions as [17]

_{p}*m*=

*n*/

_{p}*n*and where primes denote derivatives with respect to the argument. This form assumes

_{m}*=*ε ^

_{0}*x̂*, which is appropriate for our microscope. The scattering coefficients fall off rapidly with order

*n*, and the series is found to converge after a number of terms

*n*=

_{c}*ka*+ 4.05(

_{p}*ka*)

_{p}^{1/3}+ 2 [17, 18], which typically is less than 30 for micrometer-scale latex spheres in water. To compute

**f**

_{s}(

*k*

**r**) in practice, we use the accurate but numerically intensive continued fraction algorithm due to Lentz for the expansion coefficients [19

19. W. J. Lentz, “Generating Bessel functions in Mie scattering calculations using continued fractions,” Appl. Opt. **15**, 668–671 (1976). [CrossRef] [PubMed]

20. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. **19**, 1505–1509 (1980). [CrossRef] [PubMed]

*µ*m diameter polystyrene sphere in water (Polysciences Lot 526826) and a pixel-by-pixel nonlinear least-squares fit to Eqs. (7) through (10) for

**r**

_{p},

*a*and

_{p}*n*. Fits are performed with the MPFIT implementation [21] of the Levenberg-Marquardt algorithm [22] whose rigorous estimates for the adjustable parameters’ uncertainties suggest that the sphere’s center has been identified with a precision of 1 nm in all three dimensions. This estimate of the measurement resolution has been confirmed by independent measurements of similar colloidal particles’ dynamics [9, 10

_{p}10. F. C. Cheong, B. Sun, R. Dreyfus, A. Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express **17**, 13,071–13,079 (2009). [CrossRef]

*a*= 0.816 ± 0.001

_{p}*µ*m, a purely real refractive index of

*n*= 1.5821 ± 0.0006, and is centered at

_{p}*z*= 11.534 ± 0.003

_{p}*µ*m above the focal plane.

*µ*m

^{2}field of view. Wavefront curvature therefore should not affect estimates for particles’ positions.

### 3.2. Rayleigh-Sommerfeld Back-Propagation

*z*above the focal plane as the convolution,

*R*

^{2}=

*r*

^{2}+

*z*

^{2}. The sign convention for

*z*accounts for the object’s position upstream of the focal plane. Equation (12) may be rewritten with the Fourier convolution theorem as

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

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

*b*(

**r**)− 1 is

*U*

_{R}(

**q**,

*z*) is the Fourier transform of

*E*(

_{R}**r**,

*z*). From this,

*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 is reduced by defocusing, which also improves the approximations underlying Eq. (11). In the further approximation that the illumination is uniform, the reconstructed scattered field is

*I*(

_{s}**r**,

*z*) = |

*E*(

_{s}**r**,

*z*)|

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

*z*above the focal plane. The example in Fig. 3(a) shows a typical volumetric reconstruction in 0.135

*µ*m slices for a 1.5

*µ*m diameter polystyrene sphere in water, the reconstructed rays converging to the diffraction-limited bright red spot roughly 10

*µ*m above the focal plane.

*z*> 12

*µ*m in Fig. 3(a). Reconstruction artifacts overlap the images of objects that are located upstream of a scattering particle. Nevertheless, the volumetric image in Fig. 3(b) (Media 1) of the intensity scattered by a collection 1.58

*µ*m diameter silica spheres in water (Duke Scientific Lot 24169) shows clearly resolved maxima, even when spheres are occluded by their neighbors [8

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

26. E. R. Dufresne and D. G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optical elements,” Rev. Sci. Instrum. **69**, 1974–1977 (1998). [CrossRef]

*µ*m lattice constant. The colored regions in Fig. 3(b) (Media 1) are isosurfaces of the brightest 1 percent of the reconstructed voxels and clearly suggest the positions of all of the spheres in the lattice. Interpreting such a complicated hologram with fits to Lorenz-Mie theory would be challenging.

*µ*m diameter silica spheres, the second of which had a 40 nm thick cap of permalloy (80% Ni and 20% Fe) applied to roughly on octant. Asymmetric structure in the latter particle’s hologram arises from the the superposition of two contributions, the symmetric scattering pattern due to the sphere and another less well characterized contribution from its cap. Asymmetries in the hologram change form and direction as the capped sphere rotates. This structure also is apparent in the volumetric reconstructions in Figs. 4(b) and 4(c). Whereas the uncapped sphere gives rise to a rotationally symmetric intensity pattern, the capped sphere’s volumetric image has obvious structure. Both the location of the capped sphere and an indication of its orientation are captured by the volumetric reconstruction in Fig. 4, whose intensity maximum contrasts with those of ordinary spheres in that it is asymmetric.

**179**, 298–310 (1996). [CrossRef]

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

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

**179**, 298–310 (1996). [CrossRef]

*a priori*. Comparison with results of Lorenz-Mie analysis addresses this need.

## 4. Comparison of Strategies

*µ*m diameter polystyrene sphere diffusing in water sampled at 1/30 s and analyzed with both Lorenz-Mie fitting and Rayleigh-Sommerfeld back-propagation. The two approaches agree quantitatively on the particle’s inplane position, as is shown by the histogram of differences in Fig. 5(b). This distribution has peaks at 0.35 pixel offsets because brightness-weighted centroid detection is biased toward pixels’ centers unless particular care is taken to match the detection window to the size of the object [1

**179**, 298–310 (1996). [CrossRef]

*z*(

*z*

_{LM}) =

*z*

_{RS}−

*z*

_{LM}of axial positions measured with Rayleigh-Sommerfeld (RS) back propagation and Lorenz-Mie (LM) fitting as a function of

*z*

_{LM}. These differences are normally distributed about 〈Δ

*z*〉 = 1.89

*µ*m and vary only slightly with axial position.

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

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

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

**179**, 298–310 (1996). [CrossRef]

**17**, 13,071–13,079 (2009). [CrossRef]

29. P. Messmer, P. J. Mullowney, and B. E. Granger, “GPULib: GPU computing in high-level languages,” Comput. Sci. Eng. **10**, 70–73 (2008). [CrossRef]

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

**17**, 13,071–13,079 (2009). [CrossRef]

31. F. C. Cheong, S. Duarte, S.-H. Lee, and D. G. Grier, “Holographic microrheology of polysaccharides from *Streptococcus mutans* biofilms,” Rheol. Acta48, 109–115 (2009). [CrossRef]

**17**, 13,071–13,079 (2009). [CrossRef]

## 5. Conclusion

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

## Acknowledgements

## References and links

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

2. | J. C. Crocker and D. G. Grier, “Microscopic measurement of the pair interaction potential of charge-stabilized colloid,” Phys. Rev. Lett. |

3. | G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans, “Experimental demonstration of violations of the second law of thermodynmaics for small systems and short time scales,” Phys. Rev. Lett. |

4. | T. G. Mason, K. Ganesan, J. H. van Zanten, D. Wirtz, and S. C. Kuo, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. |

5. | T. T. Perkins, D. E. Smith, R. G. Larson, and S. Chu, “Stretching of a single tethered polymer in a uniform flow,” Science |

6. | C. Gosse and V. Croquette, “Magnetic tweezers: Micromanipulation and force measurement at the molecular level,” Biophys. J. |

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

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

9. | 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 |

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

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

12. | J. W. Goodman, |

13. | J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. |

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

15. | F. Dubois, L. Joannes, and J. C. Legros, “Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence,” Appl. Opt. |

16. | J. Garcia-Sucerquia, J. H. Ramírez, and R. Castaneda, “Incoherent recovering of the spatial resolution in digital holography,” Opt. Commun. |

17. | C. F. Bohren and D. R. Huffman, |

18. | M. I. Mishchenko, L. D. Travis, and A. A. Lacis, |

19. | W. J. Lentz, “Generating Bessel functions in Mie scattering calculations using continued fractions,” Appl. Opt. |

20. | W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. |

21. | C. B. Markwardt, “Non-linear least squares fitting in IDL with MPFIT,” in |

22. | J. Moré, “The Levenberg-Marquardt algorithm: Implementation and theory,” in |

23. | 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 |

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

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

26. | E. R. Dufresne and D. G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optical elements,” Rev. Sci. Instrum. |

27. | D. G. Grier, “A revolution in optical manipulation,” Nature |

28. | M. Polin, K. Ladavac, S.-H. Lee, Y. Roichman, and D. G. Grier, “Optimized holographic optical traps,” Opt. Express |

29. | P. Messmer, P. J. Mullowney, and B. E. Granger, “GPULib: GPU computing in high-level languages,” Comput. Sci. Eng. |

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

31. | F. C. Cheong, S. Duarte, S.-H. Lee, and D. G. Grier, “Holographic microrheology of polysaccharides from |

**OCIS Codes**

(090.1760) Holography : Computer holography

(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology

(180.6900) Microscopy : Three-dimensional microscopy

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: April 5, 2010

Revised Manuscript: June 2, 2010

Manuscript Accepted: June 4, 2010

Published: June 9, 2010

**Virtual Issues**

Vol. 5, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

Fook Chiong Cheong, Bhaskar Jyoti Krishnatreya, and David G. Grier, "Strategies for three-dimensional particle tracking with holographic video microscopy," Opt. Express **18**, 13563-13573 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-13-13563

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### References

- J. C. Crocker, and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179, 298–310 (1996). [CrossRef]
- J. C. Crocker, and D. G. Grier, “Microscopic measurement of the pair interaction potential of charge-stabilized colloid,” Phys. Rev. Lett. 73, 352–355 (1994). [CrossRef] [PubMed]
- G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans, “Experimental demonstration of violations of the second law of thermodynmaics for small systems and short time scales,” Phys. Rev. Lett. 89, 050601 (2002). [CrossRef] [PubMed]
- T. G. Mason, K. Ganesan, J. H. van Zanten, D. Wirtz, and S. C. Kuo, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282–3285 (1997). [CrossRef]
- T. T. Perkins, D. E. Smith, R. G. Larson, and S. Chu, “Stretching of a single tethered polymer in a uniform flow,” Science 268, 83–87 (1995). [CrossRef] [PubMed]
- C. Gosse, and V. Croquette, “Magnetic tweezers: Micromanipulation and force measurement at the molecular level,” Biophys. J. 82, 3314–3329 (2002). [CrossRef] [PubMed]
- 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, 18,275–18,282 (2007).
- F. C. Cheong, B. Sun, R. Dreyfus, A. Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express 17, 13,071–13,079 (2009). [CrossRef]
- 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]
- J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (McGraw-Hill, New York, 2005).
- J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. 45, 836–850 (2006). [CrossRef] [PubMed]
- 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]
- F. Dubois, L. Joannes, and J. C. Legros, “Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence,” Appl. Opt. 38, 7085–7094 (1999). [CrossRef]
- J. Garcia-Sucerquia, J. H. Ramírez, and R. Castaneda, “Incoherent recovering of the spatial resolution in digital holography,” Opt. Commun. 260, 62–67 (2006). [CrossRef]
- C. F. Bohren, and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Interscience, New York, 1983).
- M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University Press, Cambridge, 2001).
- W. J. Lentz, “Generating Bessel functions in Mie scattering calculations using continued fractions,” Appl. Opt. 15, 668–671 (1976). [CrossRef] [PubMed]
- W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980). [CrossRef] [PubMed]
- C. B. Markwardt, “Non-linear least squares fitting in IDL with MPFIT,” in Astronomical Data Analysis Software and Systems XVIII, D. Bohlender, P. Dowler, and D. Durand, eds. (Astronomical Society of the Pacific, San Francisco, 2009).
- J. Moré, “The Levenberg-Marquardt algorithm: Implementation and theory,” in Numerical Analysis, G. A. Watson, ed., vol. 630, p. 105 (Springer-Verlag, Berlin, 1977).
- 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]
- 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]
- E. R. Dufresne, and D. G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optical elements,” Rev. Sci. Instrum. 69, 1974–1977 (1998). [CrossRef]
- D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003). [CrossRef] [PubMed]
- M. Polin, K. Ladavac, S.-H. Lee, Y. Roichman, and D. G. Grier, “Optimized holographic optical traps,” Opt. Express 13(15), 5831–5845 (2005). [CrossRef] [PubMed]
- P. Messmer, P. J. Mullowney, and B. E. Granger, “GPULib: GPU computing in high-level languages,” Comput. Sci. Eng. 10, 70–73 (2008). [CrossRef]
- K. Xiao, and D. G. Grier, “Multidimensional optical fractionation with holographic verification,” Phys. Rev. Lett. 104, 028302 (2010). [CrossRef] [PubMed]
- F. C. Cheong, S. Duarte, S.-H. Lee, and D. G. Grier, “Holographic microrheology of polysaccharides from Streptococcus mutans biofilms,” Rheol. Acta 48, 109–115 (2009). [CrossRef]

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