## Synthetic aperture tomographic phase microscopy for 3D imaging of live cells in translational motion

Optics Express, Vol. 16, Issue 20, pp. 16240-16246 (2008)

http://dx.doi.org/10.1364/OE.16.016240

Acrobat PDF (540 KB)

### Abstract

We present a technique for 3D imaging of live cells in translational motion without need of axial scanning of objective lens. A set of transmitted electric field images of cells at successive points of transverse translation is taken with a focused beam illumination. Based on Hyugens’ principle, angular plane waves are synthesized from E-field images of a focused beam. For a set of synthesized angular plane waves, we apply a filtered back-projection algorithm and obtain 3D maps of refractive index of live cells. This technique, which we refer to as synthetic aperture tomographic phase microscopy, can potentially be combined with flow cytometry or microfluidic devices, and will enable high throughput acquisition of quantitative refractive index data from large numbers of cells.

© 2008 Optical Society of America

## 1. Introduction

2. W. Tan, A. L. Oldenburg, J. J. Norman, T. A. Desai, and S. A. Boppart, “Optical coherence tomography of cell dynamics in three-dimensional tissue models,” Opt. Express **14**, 7159–7171 (2006). [CrossRef] [PubMed]

3. J. B. Pawley, *Handbook of biological confocal microscopy* (2006). [CrossRef]

5. W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nature Methods **4**, 717–719 (2007). [CrossRef] [PubMed]

6. G. G. Levin, G. N. Vishnyakov, C. S. Zakarian, A. V. Likhachov, V. V. Pickalov, G. I. Kozinets, J. K. Novoderzhkina, and E. A. Streletskaya, “Three-dimensional limited-angle microtomography of blood cells: experimental results,” Proc. SPIE **3261**, 159–164 (1998). [CrossRef]

9. M. Zysk, J. J. Reynolds, D. L. Marks, P. S. Carney, and S. A. Boppart, “Projected index computed tomography,” Opt. Lett. **28**, 701–703 (2003). [CrossRef] [PubMed]

10. D. Nahamoo, S. X. Pan, and A. C. Kak, “Synthetic aperture diffraction tomography and its interpolation — free computer implementation,” IEEE Trans. Sonics and Ultrason. **31**, 218–229 (1984). [CrossRef]

11. T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nature Phys. **3**, 129–134 (2007). [CrossRef]

12. N. Lue, W. Choi, G. Popescu, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Confocal diffraction phase microscopy of live cells,” Opt. Lett. **33**, 2074–2076 (2008). [CrossRef] [PubMed]

13. N. Lue, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, Live cell refractometry using microfluidic devices, Opt. Lett.31, 2579 (2006). [CrossRef]

## 2. Synthetic aperture tomography theory

*A*(

*k*

*) is the complex amplitude of an individual plane wave with spatial frequency*

_{x}*k*

*. The wave number in the medium,*

_{x}*k*

_{0}=2

*π*/

*λ*

^{′}is fixed with

*λ*

^{′}=

*λ*

_{0}/

*n*

*, the wavelength in the medium;*

_{medium}*k*

*is determined by the relation*

_{z}*z*=

*z*

^{′}, can be rewritten as a weighted sum of plane waves:

*ϕ*(

*x*;

*k*

*) is a complex phase induced by a sample for each plane wave component*

_{x}*k*

*. Since these plane waves are being combined through integration, each plane wave component cannot be algebraically solved. To detangle the effect of integration, source translation along an axis perpendicular to the propagation direction is introduced. From moving the source along*

_{x}*x*-direction by

*η*, the translated plane wave can be decomposed as follows:

*ik*

_{x}*η*) is introduced as the source moves. Note that the phase shift of this additional term is dependent on

*k*

*. As a result, Fourier Transform of*

_{x}*E*(

*x*;

*η*,

*z*

^{′})at particular

*η*can be used to separate an individual spatial frequency component out of integration as follows:

*i*(

*k*

_{η}-

*k*

*)*

_{x}*η*}

*dη*=

*δ*(

*k*

_{η}-

*k*

*), eliminates the need of the integration over k*

_{x}_{x}and a complex phase

*ϕ*(

*x*;

*k*

*) of each spatial frequency component*

_{η}*k*

*can be obtained. Since an individual plane wave is retrieved by Fourier transformation, the process can also be interpreted as a synthesis of focused beams with a translation-dependent additional phase factor, exp(-*

_{η}*ik*

_{η}*η*). As a result, this method was referred to as synthetic aperture (SA) [10

10. D. Nahamoo, S. X. Pan, and A. C. Kak, “Synthetic aperture diffraction tomography and its interpolation — free computer implementation,” IEEE Trans. Sonics and Ultrason. **31**, 218–229 (1984). [CrossRef]

*k*

*=*

_{η}*k*

_{0}sin

*θ*, with

*θ*the propagation direction of plane wave relative to the optic axis, the angular phase image, φ(

*x*;

*θ*), can be determined. By interpreting the phase image as an integration of the refractive index along the beam propagation direction, a filtered back-projection algorithm can be used to obtain the 3-D map of refractive index [5

5. W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nature Methods **4**, 717–719 (2007). [CrossRef] [PubMed]

7. F. Charriere, A. Marian, F. Montfort, J. Kuehn, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Cell refractive index tomography by digital holographic microscopy,” Opt. Lett. **31**, 178–180 (2006). [CrossRef] [PubMed]

## 3. Experimental setup

15. C. Fang-Yen, S. Oh, Y. Park, W. Choi, S. Song, H. S. Seung, R. R. Dasari, and M. S. Feld, “Imaging voltage-dependent cell motions with heterodyne Mach-Zehnder phase microscopy,” Opt. Lett. **32**, 1572–1574 (2007). [CrossRef] [PubMed]

10. D. Nahamoo, S. X. Pan, and A. C. Kak, “Synthetic aperture diffraction tomography and its interpolation — free computer implementation,” IEEE Trans. Sonics and Ultrason. **31**, 218–229 (1984). [CrossRef]

*k*

*) and the Imaging axis in spatial coordinate (*

_{x}*y*) at the camera plane. The reference beam, a planar beam whose frequency is shifted by 1.25kHz using two acousto-optic modulators as mentioned previously [5

5. W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nature Methods **4**, 717–719 (2007). [CrossRef] [PubMed]

15. C. Fang-Yen, S. Oh, Y. Park, W. Choi, S. Song, H. S. Seung, R. R. Dasari, and M. S. Feld, “Imaging voltage-dependent cell motions with heterodyne Mach-Zehnder phase microscopy,” Opt. Lett. **32**, 1572–1574 (2007). [CrossRef] [PubMed]

*ϕ*(

*k*

*,*

_{x}*y*) is obtained (Fig. 1(f)). To generate a scanning point source for this synthetic aperture tomography, the sample is translated across the line focus beam, along the Fourier axis by a precision micro-position translation stage (PI M-216, Physik Instrumente Germany) with a step size of 0.1 µm.

## 4. Data acquisition and processing

*ϕ*(

*k*

*,*

_{x}*y*;

*η*), taken as a function of sample translation

*η*is shown in Fig. 2(a). The translation range is typically greater than twice the diameter of the sample. Corresponding amplitude images,

*A*(

*k*

*,*

_{x}*y*;

*η*), can also be obtained from the 4f-PSI algorithm. After combining both amplitude and phase to obtain the electric field,

*E*(

*k*

*,*

_{x}*y*;

*η*)=

*Ae*

*, a numerical inverse Fourier transform is carried out along the*

^{iϕ}*k*

*-axis to obtain*

_{x}*E*(

*x*;

*η*,

*y*). Figure 2(b) shows the resulting amplitude image in logarithmic scale. The illumination beam is stationary in the experiment and the sample is moving. In order to employ the synthetic aperture method, the sample is set to be stationary while the illumination beam is translated. Using known translation

*η*from the translation stage, we numerically shift the image in the opposite direction of translation, which is equivalent to shifting the focused beam along x-direction while the sample is stationary.

*E*(

*x*;

*η*,

*y*) taken for the sample in translational motion, we applied synthetic aperture algorithm described in the theory section. For any given y, we take Fourier transform of

*E*(

*x*;

*η*,

*y*) for the sample translation

*η*as described in Eq. (4) as follows:

*φ*(

*x*,

*y*;

*k*

*) induced by the sample can be obtained (Fig. 2(c)). Using the relationship*

_{η}*k*

*=*

_{η}*k*

_{0}sin

*θ*, with

*θ*the direction of plane wave relative to the optic axis, the angular projection phase image,

*φ*(

*x*,

*y*;

*θ*), can be determined.

*φ*(

*x*,

*y*;

*θ*) for a HeLa cell at angles (

*θ*): -30, 0 and +30 degrees, respectively. Note that the phase image for nonzero degree is elongated along the tilting direction (x-axis). This is due to the fixed image plane during tilting illumination. A set of angular projection phase images ranging from -40 to 40 degrees can be synthesized from this system. After applying the filtered back-projection algorithm, the 3D map of refractive index is obtained by applying an inverse radon transform [17].

8. V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. **205**, 165–176 (2002). [CrossRef] [PubMed]

**4**, 717–719 (2007). [CrossRef] [PubMed]

**4**, 717–719 (2007). [CrossRef] [PubMed]

## 5. Summary and conclusions

## Acknowledgments

## References and links

1. | B. E. Bouma and G. J. Tearney, |

2. | W. Tan, A. L. Oldenburg, J. J. Norman, T. A. Desai, and S. A. Boppart, “Optical coherence tomography of cell dynamics in three-dimensional tissue models,” Opt. Express |

3. | J. B. Pawley, |

4. | M. Slaney and A. C. Kak, “Diffraction tomography,” Proc. Soc. Photo. Opt. Instrum. Eng. |

5. | W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nature Methods |

6. | G. G. Levin, G. N. Vishnyakov, C. S. Zakarian, A. V. Likhachov, V. V. Pickalov, G. I. Kozinets, J. K. Novoderzhkina, and E. A. Streletskaya, “Three-dimensional limited-angle microtomography of blood cells: experimental results,” Proc. SPIE |

7. | F. Charriere, A. Marian, F. Montfort, J. Kuehn, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Cell refractive index tomography by digital holographic microscopy,” Opt. Lett. |

8. | V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. |

9. | M. Zysk, J. J. Reynolds, D. L. Marks, P. S. Carney, and S. A. Boppart, “Projected index computed tomography,” Opt. Lett. |

10. | D. Nahamoo, S. X. Pan, and A. C. Kak, “Synthetic aperture diffraction tomography and its interpolation — free computer implementation,” IEEE Trans. Sonics and Ultrason. |

11. | T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nature Phys. |

12. | N. Lue, W. Choi, G. Popescu, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Confocal diffraction phase microscopy of live cells,” Opt. Lett. |

13. | N. Lue, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, Live cell refractometry using microfluidic devices, Opt. Lett.31, 2579 (2006). [CrossRef] |

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

15. | C. Fang-Yen, S. Oh, Y. Park, W. Choi, S. Song, H. S. Seung, R. R. Dasari, and M. S. Feld, “Imaging voltage-dependent cell motions with heterodyne Mach-Zehnder phase microscopy,” Opt. Lett. |

16. | X. Zhong, “A four-frame phase shift method insensitive to phase shifter nonlinearity,” J. Opt. A: Pure Appl. Opt. |

17. | A. C. Kak and M. Slaney, |

**OCIS Codes**

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(170.3880) Medical optics and biotechnology : Medical and biological imaging

(180.0180) Microscopy : Microscopy

**ToC Category:**

Microscopy

**History**

Original Manuscript: August 1, 2008

Revised Manuscript: September 19, 2008

Manuscript Accepted: September 24, 2008

Published: September 26, 2008

**Virtual Issues**

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

**Citation**

Niyom Lue, Wonshik Choi, Gabriel Popescu, Kamran Badizadegan, Ramachandra R. Dasari, and Michael S. Feld, "Synthetic aperture tomographic phase microscopy for 3D imaging of live cells in translational motion," Opt. Express **16**, 16240-16246 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-16240

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### References

- B. E. Bouma and G. J. Tearney, Handbook of optical coherent tomography (2001).
- W. Tan, A. L. Oldenburg, J. J. Norman, T. A. Desai, and S. A. Boppart, "Optical coherence tomography of cell dynamics in three-dimensional tissue models," Opt. Express 14, 7159-7171 (2006). [CrossRef] [PubMed]
- J. B. Pawley, Handbook of biological confocal microscopy (2006). [CrossRef]
- M. Slaney and A. C. Kak, "Diffraction tomography," Proc. Soc. Photo. Opt. Instrum. Eng. 413, 2-19 (1983).
- W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic phase microscopy," Nature Methods 4, 717-719 (2007). [CrossRef] [PubMed]
- G. G. Levin, G. N. Vishnyakov, C. S. Zakarian, A. V. Likhachov, V. V. Pickalov, G. I. Kozinets, J. K. Novoderzhkina, and E. A. Streletskaya, "Three-dimensional limited-angle microtomography of blood cells: experimental results," Proc. SPIE 3261, 159-164 (1998). [CrossRef]
- F. Charriere, A. Marian, F. Montfort, J. Kuehn, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, "Cell refractive index tomography by digital holographic microscopy," Opt. Lett. 31, 178-180 (2006). [CrossRef] [PubMed]
- V. Lauer, "New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope," J. Microsc. 205, 165-176 (2002). [CrossRef] [PubMed]
- M. Zysk, J. J. Reynolds, D. L. Marks, P. S. Carney, and S. A. Boppart, "Projected index computed tomography," Opt. Lett. 28, 701-703 (2003). [CrossRef] [PubMed]
- D. Nahamoo, S. X. Pan, and A. C. Kak, "Synthetic aperture diffraction tomography and its interpolation -free computer implementation," IEEE Trans. Sonics and Ultrason. 31, 218-229 (1984). [CrossRef]
- T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, "Interferometric synthetic aperture microscopy," Nature Phys. 3, 129-134 (2007). [CrossRef]
- N. Lue, W. Choi, G. Popescu, K. Badizadegan, R. R. Dasari, and M. S. Feld, "Confocal diffraction phase microscopy of live cells," Opt. Lett. 33, 2074-2076 (2008). [CrossRef] [PubMed]
- N. Lue, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, Live cell refractometry using microfluidic devices, Opt. Lett. 31, 2579 (2006). [CrossRef]
- M. Born and E. Wolf, Principles of optics (1999).
- C. Fang-Yen, S. Oh, Y. Park, W. Choi, S. Song, H. S. Seung, R. R. Dasari, and M. S. Feld, "Imaging voltage-dependent cell motions with heterodyne Mach-Zehnder phase microscopy," Opt. Lett. 32, 1572-1574 (2007). [CrossRef] [PubMed]
- X. Zhong, "A four-frame phase shift method insensitive to phase shifter nonlinearity," J. Opt. A: Pure Appl. Opt. 8, 300-303 (2006).
- A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (1988).

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