## 3D Localization of weak scatterers in digital holographic microscopy using Rayleigh-Sommerfeld back-propagation |

Optics Express, Vol. 20, Issue 15, pp. 16735-16744 (2012)

http://dx.doi.org/10.1364/OE.20.016735

Acrobat PDF (1118 KB)

### Abstract

The Rayleigh-Sommerfeld back-propagation method is a fast and highly flexible volume reconstruction scheme for digital holographic microscopy. We present a new method for 3D localization of weakly scattering objects using this technique. A well-known aspect of classical optics (the Gouy phase shift) can be used to discriminate between objects lying on either side of the holographic image plane. This results in an unambiguous, model-free measurement of the axial coordinate of microscopic samples, and is demonstrated both on an individual colloidal sphere, and on a more complex object — a layer of such particles in close contact.

© 2012 OSA

## 1. Introduction

1. S. 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]

2. J. Fung, K. E. Martin, R. W. Perry, D. M. Katz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Opt. Express **19**, 8051–8065 (2011). [CrossRef] [PubMed]

3. W. S. Haddad, D. Cullen, J. C. Solem, J. W. Longworth, A. McPherson, K. Boyer, and C. K. Rhodes, “Fourier-transform holographic microscope,” Appl. Opt. **31**, 4973–4978 (1992). [CrossRef] [PubMed]

6. Z. Frentz, S. Kuehn, D. Hekstra, and S. Leiber, “Microbial population dynamics by digital in-line holographic microscopy,” Rev. Sci. Inst. **81**, 084301 (2010). [CrossRef]

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

8. R. Besseling, E. Weeks, A. Schofield, and W. Poon, “Three-dimensional imaging of colloidal glasses under steady shear,” Phys. Rev. Lett. **99**, 028301 (2007). [CrossRef] [PubMed]

9. J. Conrad, M. Gibiansky, F. Jin, V. Gordon, D. Motto, M. Mathewson, W. Stopka, D. Zelasko, J. Shrout, and G. Wong, “Flagella and pili-mediated near-surface single-cell motility mechanisms in p. aeruginosa,” Biophys. J. **100**, 1608–1616 (2011). [CrossRef] [PubMed]

10. A. van Blaaderen and P. Wiltzius, “Real-space structure of colloidal hard-sphere glasses,” Science **270**, 1177–1179 (1995). [CrossRef]

1. S. 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]

2. J. Fung, K. E. Martin, R. W. Perry, D. M. Katz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Opt. Express **19**, 8051–8065 (2011). [CrossRef] [PubMed]

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

13. G. Pan and H. Meng, “Digital holography of particle fields: reconstruction by use of complex amplitude,” Appl. Opt. **42**, 827–833 (2003). [CrossRef] [PubMed]

15. 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]

5. J. Sheng, E. Malkiel, J. Katz, J. Adolf, R. B. amnd, and A. R. Place, “Digital holographic microscopy reveals prey-induced changes in swimming behavior of predatory dinoflagellates,” Proc. Natl. Acad. Sci. USA **104**, 17512–17517 (2007). [CrossRef] [PubMed]

## 2. Theory and methods

*a*, in a typical DHM setup (Fig. 1(b)). A plane wave with amplitude

*E*

_{0}is incident on a particle located at the origin. The scattered field

**E**

*interferes with the unscattered light*

_{s}**E**

_{0}, resulting in a total intensity at a point

**r**[29

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

**E**

*(*

_{s}**r**)|

^{2}is small, and so can be ignored. Dividing through by the unscattered intensity we obtain the ‘normalized hologram’

*b*(

**r**): Each dipole element in the scatterer gives a separate contribution to the scattered field; these contributions may be integrated over the volume of the scatterer to give the total scattered field [19]. The incident, unpolarized field may be represented as the superposition of two orthogonally polarized beams. This leads to a scattering pattern which is azimuthally symmetric about the

*z*-axis, where

*V*is the volume of the scatterer, and

*f*(

*θ*) is the form factor for a sphere [20], Throughout, we assume that the scatterer is far enough from the scattering plane that geometric rotation of polarization is small [29

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

*μ*m from the scatterer (i.e. the scatterer was located at

*z*= −40

_{p}*μ*m in our coordinate system, as we consider the center of the hologram plane to be the origin), and measures 512 pixels in each dimension. The other relevant parameters were

*λ*= 0.505

*μ*m,

*a*= 0.1

*μ*m, the particle and surrounding medium refractive indices

*n*= 1.55 and

_{p}*n*= 1.33 respectively, and the sampling frequency in the

_{m}*x*,

*y*plane, 10 pixels/

*μ*m. Based on this simulated data, we reconstruct the light field associated with the particle using the Rayleigh-Sommerfeld scheme. This method has been described in detail elsewhere [11, 12

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

*z*′ with the use of the Rayleigh-Sommerfeld propagator where

*r*′ = (

*x*′

^{2}+

*y*′

^{2}+

*z*′

^{2})

^{1/2}. Note the use of primed coordinates to indicate a position in the reconstructed (as opposed to physical) volume. We can reconstruct the field (and from there, the image) at any height above the hologram plane by the convolution To demonstrate this, a stack of 512 images were numerically reconstructed, at a spacing along the optical axis of Δ

*z*′ = −0.156

*μ*m, where the sign of this increment indicates a reconstruction upstream (from hologram to scatterer).

*y*′ = 0 (i.e. through the center of the scatterer) is shown in Fig. 2(a). Note the small bright region below the midpoint and the small dark region just above the midpoint; this feature is due to the phase shift on passing through the geometrical focus. This image was then convolved with a kernel based on the Sobel filter [30] to extract the intensity gradient in the negative

*z*′-direction only, which we denote

*g*(

*x*′,

*z*′): where the filter kernel

*S*′

*is given by For the approximation in Eq. (8) to be valid, the optical field must vary slowly compared to the sampling frequency in the axial direction. This condition is easily satisfied, as Δ*

_{z}*z*′ can be made arbitrarily small. For clarity, we retain only the values of

*g*(

*x*′,

*z*′) greater than zero, setting all negative values to zero. The result of this process can be seen in Fig. 2(b). This gradient operation provides a very efficient method of locating the center of the scatterer, with sub-pixel resolution. To demonstrate this, we extracted the intensity profiles along lines through the center vertical (red) and horizontal (black) directions of each of the images. These plots are shown in Fig. 2(c) and Fig. 2(d). In both panels, the axial intensity profile is shifted by +40

*μ*m, to place the scatterer’s center at the origin. Note also that this method has yielded the center position of a scatterer whose

*z*-coordinate is negative; the field, imaged onto the camera sensor is

*diverging*. If a particle’s

*z*-coordinate is positive, the field is

*converging*when it arrives at the image sensor and has not yet passed through its geometrical focus. This allows us to separate objects lying on different sides of the hologram plane unambiguously. Whether the particles have a physical position

*z*above or below the focal plane, they have twin images at ±

_{p}*z*′

*in the reconstructed space. However, the twin images are*

_{p}*not*identical. If we only reconstruct the half-space ‘upstream’ of the focal plane (i.e.

*z*′ < 0), we find that

*g*(

*z*′

*) is a maximum for particles with*

_{p}*z*< 0, and a minimum for those with

*z*> 0. As a check of this method, we also generated a hologram using the more accurate Mie scattering solution for the scattered field [19, 29

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

*z*= −39.92

*μ*m, an error of 0.2%.

*λ*= 0.505

*μ*m LED (Thorlabs M505L2) placed directly behind the condenser iris. The iris itself was closed as far as possible, with a remaining clear aperture of ∼ 1.5mm. The resulting quasi-plane wave illumination in the sample plane was sufficient to generate the interference rings seen in the defocused particle images (see results, Fig. 3). Although collimated lasers are often used as a light source [6

6. Z. Frentz, S. Kuehn, D. Hekstra, and S. Leiber, “Microbial population dynamics by digital in-line holographic microscopy,” Rev. Sci. Inst. **81**, 084301 (2010). [CrossRef]

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

15. 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]

31. L. Repetto, E. Piano, and C. Pontiggia, “Lensless digital holographic microscope with light-emitting diode illumination,” Opt. Lett. **29**, 1132–1134 (2004). [CrossRef] [PubMed]

*μ*m either side of the the focal plane; objects outside this volume are effectively invisible.

*n*= 1.515 at 23°C). This was done to minimize the effects of aberrations incurred when imaging through water into glass and to limit any multiple reflections between the particles and the wall of the sample chamber. The stepper motor used to position the objective lens turret was used to defocus the image of a single isolated particle by a defined amount. Although the calibrated motor readout implied a precision of ±25 nm, this did not accurately reflect our ability to reproducibly place a particle in ‘perfect focus’. The slight flexibility of the sample chamber wall and the viscosity of the immersion oil (which causes the coverslip to deform as the objective lens is moved) place a more conservative uncertainty on the true

_{d}*z*position of the particle. Furthermore, the selection of a perfectly in-focus plane (i.e. placing a particle at

*z*= 0

*μ*m) was performed by eye, and was hence a little subjective. Based on several cycles of defocusing then refocusing the image of a layer of particles, we estimate the total uncertainty in

*z*to be ±150 nm.

*μ*m across a range of defocus, giving a range of axial particle coordinates from

*z*= −50

_{p}*μ*m to +50

*μ*m (

*z*= 0

_{p}*μ*m implies that the particle lies in the focal plane). Next, a patch of closely-spaced particles was found. Images of this particle layer were acquired at different defocus distances. This second experiment demonstrates the technique’s use for measuring the

*z*-coordinate of a flat, extended object. In both cases, the filter

*S*

_{z}_{′}was convolved with every reconstructed

*x*′

*z*′ plane. Due to the chosen form of the convolution kernel, an identical stack of gradient images

*g*(

*x*′,

*y*′,

*z*′) would be obtained by convolution with the

*y*′

*z*′ planes. In both the single- and multiple-particle cases, images were taken with and without the sample in place. This allowed the removal of unwanted background artifacts caused by imperfections in the optical system [12

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

## 3. Results

*xz*slice through a stack of images of a single particle, created by manually refocusing the stage at Δ

*z*= 1

*μ*m intervals (a similar process was implemented by Elliot and Poon [34

34. M. Elliot and W. Poon, “Conventional optical microscopy of colloidal suspensions,” Adv. Coll. Interf. Sci. **92**, 133–194 (2001). [CrossRef]

*z*′ =

*z*= 0), this would be a map of the intensity close to the camera’s image sensor. Figure 3(b) and Fig. 3(e) show example

*xy*images of an isolated particle at

*z*= 9

*μ*m and

*z*= −9

*μ*m respectively. From each of these images, a stack similar to that in Fig. 3(a) can be reconstructed (Figs. 3(c) and 3(f)), using Eq.s (3) and (7).

*x*′

*z*′ images, but the important qualitative features are the same. In both cases, the particle is located in the center of the panel. Figure 3(d) shows

*g*(

*x*′,

*z*′) < 0 from Fig. 3(b), and Fig. 3(g) shows

*g*(

*x*′,

*z*′) > 0 from Fig. 3(f), clearly showing the position of the particle in both cases. These intensity features may be fitted in three dimensions in order to accurately localize the particle. We find

*z*= 8.9±0.15

*μ*m for Fig. 3(d) and

*z*= −9.1±0.15

*μ*m for Fig. 3(g), with uncertainties dominated by the ±150 nm instrumental error.

*z*= 0

*μ*m to

*z*= −80

*μ*m in steps of Δ

*z*= −4

*μ*m. Similarly accurate results were obtained from data in the range

*z*= 0

*μ*m → 80

*μ*m (data not shown). From each of these stage displacements, a full stack of images was calculated, and from these we obtained the gradient

*g*(

*x*′,

*y*′,

*z*′) as before. As opposed to fitting subregions corresponding to individual particles, we integrated over all

*x*′,

*y*′ values at constant

*z*′, for each actual stage displacement

*z*. This yielded a single value which is a function of actual (

*z*) and reconstructed (

*z*′) defocus, and which we denote

*G*(

*z*,

*z*′) = ∫ (

*g z;x*′,

*y*′,

*z*′)d

*x*′d

*y*′. These one-dimensional quantities are plotted in Fig. 4, where each row of colors corresponds to an actual defocus distance

*z*. The horizontal axis shows reconstructed defocus distance, and the colors indicate the magnitude of

*G*(

*z*,

*z*′). The data have been rescaled such that the maximum value of

*G*(

*z*,

*z*′) is unity for each actual defocus

*z*. Although the peaks in

*G*(

*z*,

*z*′) broaden considerably as a function of

*z*, the peak values are well-defined and easy to fit in all cases.

## 4. Conclusion

## References and links

1. | S. 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 |

2. | J. Fung, K. E. Martin, R. W. Perry, D. M. Katz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Opt. Express |

3. | W. S. Haddad, D. Cullen, J. C. Solem, J. W. Longworth, A. McPherson, K. Boyer, and C. K. Rhodes, “Fourier-transform holographic microscope,” Appl. Opt. |

4. | W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. USA |

5. | J. Sheng, E. Malkiel, J. Katz, J. Adolf, R. B. amnd, and A. R. Place, “Digital holographic microscopy reveals prey-induced changes in swimming behavior of predatory dinoflagellates,” Proc. Natl. Acad. Sci. USA |

6. | Z. Frentz, S. Kuehn, D. Hekstra, and S. Leiber, “Microbial population dynamics by digital in-line holographic microscopy,” Rev. Sci. Inst. |

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

8. | R. Besseling, E. Weeks, A. Schofield, and W. Poon, “Three-dimensional imaging of colloidal glasses under steady shear,” Phys. Rev. Lett. |

9. | J. Conrad, M. Gibiansky, F. Jin, V. Gordon, D. Motto, M. Mathewson, W. Stopka, D. Zelasko, J. Shrout, and G. Wong, “Flagella and pili-mediated near-surface single-cell motility mechanisms in p. aeruginosa,” Biophys. J. |

10. | A. van Blaaderen and P. Wiltzius, “Real-space structure of colloidal hard-sphere glasses,” Science |

11. | J. W. Goodman, |

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

13. | G. Pan and H. Meng, “Digital holography of particle fields: reconstruction by use of complex amplitude,” Appl. Opt. |

14. | C. Fournier, C. Ducottet, and T. Fournel, “Digital in-line holography: influence of the reconstruction function on the axial profile of a reconstructed particle image,” Meas. Sci. Technol. |

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

16. | M. Kim, “Principles and techniques of digital holographic microscopy,” SPIE Rev. |

17. | J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Ann. Rev. Fluid Mech. |

18. | D. Gabor, “A new microscopic principle,” Nature |

19. | C. Bohren and D. Huffman, |

20. | B. Berne and R. Pecora, |

21. | U. Agero, C. Monken, C. Ropert, R. Gazzinelli, and O. Mesquita, “Cell surface fluctuations studied with defocusing microscopy,” Phys. Rev. E |

22. | L. Mesquita, U. Agero, and O. Mesquita, “Defocusing microscopy: An approach for red blood cell optics,” Appl. Phys. Lett. |

23. | L. Waller, L. Tian, and G. Barbastathis, “Transport of intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express |

24. | M. Born and E. Wolf, |

25. | G. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can. J. Phys. |

26. | A. Pralle, M. Prummer, E.-L. Florin, E. Stelzer, and J. Hörber, “Three-dimensional high-resolution particle tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. |

27. | A. Rohrbach and E. Stelzer, “Three-dimensional position detection of optically trapped dielectric particles,” J. Appl. Phys. |

28. | L. Wilson, A. Harrison, A. Schofield, J. Arlt, and W. Poon, “Passive and active microrheology of hard-sphere colloids,” J. Phys. Chem. B |

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

30. | E. L. Hall, |

31. | L. Repetto, E. Piano, and C. Pontiggia, “Lensless digital holographic microscope with light-emitting diode illumination,” Opt. Lett. |

32. | B. Kemper, S. Stürwald, C. Remmersmann, P. Langehanenberg, and G. von Bally, “Characterisation of light emitting diodes (leds) for application in digital holographic microscopy for inspection of micro and nanostructured surfaces,” Opt. Laser. Eng. |

33. | B. Kemper, S. Kosmeier, P. Langehanenberg, S. Przibilla, C. Remmersmann, S. Stürwald, and G. von Bally, “Application of 3d tracking, led illumination and multi-wavelength techniques for quantitative cell analysis in digital holographic microscopy,” Proc. SPIE |

34. | M. Elliot and W. Poon, “Conventional optical microscopy of colloidal suspensions,” Adv. Coll. Interf. Sci. |

**OCIS Codes**

(180.6900) Microscopy : Three-dimensional microscopy

(090.1995) Holography : Digital holography

**ToC Category:**

Microscopy

**History**

Original Manuscript: June 18, 2012

Revised Manuscript: June 28, 2012

Manuscript Accepted: June 28, 2012

Published: July 9, 2012

**Virtual Issues**

Vol. 7, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Laurence Wilson and Rongjing Zhang, "3D Localization of weak scatterers in digital holographic microscopy using Rayleigh-Sommerfeld back-propagation," Opt. Express **20**, 16735-16744 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-15-16735

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### References

- S. 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. Express15, 18275–18282 (2007). [CrossRef] [PubMed]
- J. Fung, K. E. Martin, R. W. Perry, D. M. Katz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Opt. Express19, 8051–8065 (2011). [CrossRef] [PubMed]
- W. S. Haddad, D. Cullen, J. C. Solem, J. W. Longworth, A. McPherson, K. Boyer, and C. K. Rhodes, “Fourier-transform holographic microscope,” Appl. Opt.31, 4973–4978 (1992). [CrossRef] [PubMed]
- W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. USA98, 11301–11305 (2001). [CrossRef] [PubMed]
- J. Sheng, E. Malkiel, J. Katz, J. Adolf, R. B. amnd, and A. R. Place, “Digital holographic microscopy reveals prey-induced changes in swimming behavior of predatory dinoflagellates,” Proc. Natl. Acad. Sci. USA104, 17512–17517 (2007). [CrossRef] [PubMed]
- Z. Frentz, S. Kuehn, D. Hekstra, and S. Leiber, “Microbial population dynamics by digital in-line holographic microscopy,” Rev. Sci. Inst.81, 084301 (2010). [CrossRef]
- J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci.179, 298–310 (1996). [CrossRef]
- R. Besseling, E. Weeks, A. Schofield, and W. Poon, “Three-dimensional imaging of colloidal glasses under steady shear,” Phys. Rev. Lett.99, 028301 (2007). [CrossRef] [PubMed]
- J. Conrad, M. Gibiansky, F. Jin, V. Gordon, D. Motto, M. Mathewson, W. Stopka, D. Zelasko, J. Shrout, and G. Wong, “Flagella and pili-mediated near-surface single-cell motility mechanisms in p. aeruginosa,” Biophys. J.100, 1608–1616 (2011). [CrossRef] [PubMed]
- A. van Blaaderen and P. Wiltzius, “Real-space structure of colloidal hard-sphere glasses,” Science270, 1177–1179 (1995). [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics3rd Ed., (Roberts & Company, 2005).
- S.-H. Lee and D. G. Grier, “Holographic microscopy of holographically trapped three-dimensional structures,” Opt. Express15, 1505–1512 (2007). [CrossRef] [PubMed]
- G. Pan and H. Meng, “Digital holography of particle fields: reconstruction by use of complex amplitude,” Appl. Opt.42, 827–833 (2003). [CrossRef] [PubMed]
- C. Fournier, C. Ducottet, and T. Fournel, “Digital in-line holography: influence of the reconstruction function on the axial profile of a reconstructed particle image,” Meas. Sci. Technol.15, 686–693 (2004). [CrossRef]
- 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]
- M. Kim, “Principles and techniques of digital holographic microscopy,” SPIE Rev.1, 018005 (2010). [CrossRef]
- J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Ann. Rev. Fluid Mech.42, 531–555 (2010). [CrossRef]
- D. Gabor, “A new microscopic principle,” Nature161, 18275–18282 (1948). [CrossRef]
- C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons., 1983).
- B. Berne and R. Pecora, Dynamic Light Scattering (John Wiley & Sons, Inc., 1976).
- U. Agero, C. Monken, C. Ropert, R. Gazzinelli, and O. Mesquita, “Cell surface fluctuations studied with defocusing microscopy,” Phys. Rev. E67, 051904 (2003). [CrossRef]
- L. Mesquita, U. Agero, and O. Mesquita, “Defocusing microscopy: An approach for red blood cell optics,” Appl. Phys. Lett.88, 133901 (2006). [CrossRef]
- L. Waller, L. Tian, and G. Barbastathis, “Transport of intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express18, 12552–12561 (2010). [CrossRef] [PubMed]
- M. Born and E. Wolf, Principles of Optics6th Ed., (Cambridge University Press, 1998).
- G. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can. J. Phys.36, 935–943 (1958). [CrossRef]
- A. Pralle, M. Prummer, E.-L. Florin, E. Stelzer, and J. Hörber, “Three-dimensional high-resolution particle tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech.44, 378–86 (1999). [CrossRef] [PubMed]
- A. Rohrbach and E. Stelzer, “Three-dimensional position detection of optically trapped dielectric particles,” J. Appl. Phys.91, 5474 (2002). [CrossRef]
- L. Wilson, A. Harrison, A. Schofield, J. Arlt, and W. Poon, “Passive and active microrheology of hard-sphere colloids,” J. Phys. Chem. B113, 3806–3812 (2009). [CrossRef] [PubMed]
- F. Cheong, B. Krishnatreya, and D. Grier, “Strategies for three-dimensional particle tracking with holographic video microscopy,” Opt. Express18, 13563–13573 (2010). [CrossRef] [PubMed]
- E. L. Hall, Computer Image Processing and Recognition (Academic Press, 1979).
- L. Repetto, E. Piano, and C. Pontiggia, “Lensless digital holographic microscope with light-emitting diode illumination,” Opt. Lett.29, 1132–1134 (2004). [CrossRef] [PubMed]
- B. Kemper, S. Stürwald, C. Remmersmann, P. Langehanenberg, and G. von Bally, “Characterisation of light emitting diodes (leds) for application in digital holographic microscopy for inspection of micro and nanostructured surfaces,” Opt. Laser. Eng.46, 499–507 (2008). [CrossRef]
- B. Kemper, S. Kosmeier, P. Langehanenberg, S. Przibilla, C. Remmersmann, S. Stürwald, and G. von Bally, “Application of 3d tracking, led illumination and multi-wavelength techniques for quantitative cell analysis in digital holographic microscopy,” Proc. SPIE7184, 71840R–1–71840R–12 (2009).
- M. Elliot and W. Poon, “Conventional optical microscopy of colloidal suspensions,” Adv. Coll. Interf. Sci.92, 133–194 (2001). [CrossRef]

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