## Simplified approach to diffraction tomography in optical microscopy

Optics Express, Vol. 17, Issue 15, pp. 12407-12417 (2009)

http://dx.doi.org/10.1364/OE.17.012407

Acrobat PDF (684 KB)

### Abstract

We present a novel microscopy technique to measure the scattered wavefront emitted from an optically transparent microscopic object. The complex amplitude is decoded via phase stepping in a common-path interferometer, enabling high mechanical stability. We demonstrate theoretically and practically that the incoherent summation of multiple illumination directions into a single image increases the resolving power and facilitates image reconstruction in diffraction tomography. We propose a slice-by-slice object-scatter extraction algorithm entirely based in real space in combination with ordinary z-stepping. Thereby the computational complexity affiliated with tomographic methods is significantly reduced. Using the first order Born approximation for weakly scattering objects it is possible to obtain estimates of the scattering density from the exitwaves.

© 2009 OSA

## 1. Introduction

1. F. Zernike, “How I discovered phase contrast,” Science **121**(3141), 345–349 (1955). [CrossRef] [PubMed]

2. R. D. Allen, G. B. David, and G. Nomarski, “The zeiss-Nomarski differential interference equipment for transmitted-light microscopy,” Z. Wiss. Mikrosk. **69**(4), 193–221 (1969). [PubMed]

3. O. Shimomura, “The discovery of aequorin and green fluorescent protein,” J. Microsc. **217**(1), 3–15 (2005). [CrossRef]

4. L. Liu, J. R. Trimarchi, R. Oldenbourg, and D. L. Keefe, “Increased birefringence in the meiotic spindle provides a new marker for the onset of activation in living oocytes,” Biol. Reprod. **63**(1), 251–258 (2000). [CrossRef] [PubMed]

5. M. Shribak and S. Inoué, “Orientation-independent differential interference contrast microscopy,” Appl. Opt. **45**(3), 460–469 (2006). [CrossRef] [PubMed]

6. N. Lue, W. Choi, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, “Quantitative phase imaging of live cells using fast Fourier phase microscopy,” Appl. Opt. **46**(10), 1836–1842 (2007). [CrossRef] [PubMed]

7. G. Popescu, L. P. Deflores, J. C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, and M. S. Feld, “Fourier phase microscopy for investigation of biological structures and dynamics,” Opt. Lett. **29**(21), 2503–2505 (2004). [CrossRef] [PubMed]

8. E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holography for quantitative phase-contrast imaging,” Opt. Lett. **24**(5), 291–293 (1999). [CrossRef]

10. U. Schnars and W. P. Juptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. **33**(2), 179–181 (1994). [CrossRef] [PubMed]

11. F. Charrière, J. Kühn, T. Colomb, F. Montfort, E. Cuche, Y. Emery, K. Weible, P. Marquet, and C. Depeursinge, “Characterization of microlenses by digital holographic microscopy,” Appl. Opt. **45**(5), 829–835 (2006). [CrossRef] [PubMed]

14. B. Kemper, D. Carl, J. Schnekenburger, I. Bredebusch, M. Schafer, W. Domschke, and G. von Bally, “Investigation of living pancreas tumor cells by digital holographic microscopy,” J. Biomed. Opt. **11**(3), 034005–034008 (2006). [CrossRef]

15. S. S. Kou and C. J. Sheppard, “Imaging in digital holographic microscopy,” Opt. Express **15**(21), 13640–13648 (2007). [CrossRef] [PubMed]

16. F. Charrière, 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**(2), 178–180 (2006). [CrossRef] [PubMed]

21. V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. **205**(2), 165–176 (2002). [CrossRef] [PubMed]

16. F. Charrière, 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**(2), 178–180 (2006). [CrossRef] [PubMed]

17. W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods **4**(9), 717–719 (2007). [CrossRef] [PubMed]

18. M. Debailleul, B. Simon, V. Georges, O. Haeberlé, and V. Lauer, “Holographic microscopy and diffractive microtomography of transparent samples,” Meas. Sci. Technol. **19**(7), 074009 (2008). [CrossRef]

21. V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. **205**(2), 165–176 (2002). [CrossRef] [PubMed]

18. M. Debailleul, B. Simon, V. Georges, O. Haeberlé, and V. Lauer, “Holographic microscopy and diffractive microtomography of transparent samples,” Meas. Sci. Technol. **19**(7), 074009 (2008). [CrossRef]

## 2. Theory

*K*is the spatial frequency of the object,

*k*is the wave vector of the diffracted wave and

*k*is the wave vector of the illumination wave. By geometrical construction one can show that all the observable object information lies on a sphere that is centered at

_{i}*-k*in reciprocal space [22]. Commonly this sphere is called the Ewald sphere in X-ray diffraction. Wolf first proposed that one could use varying illumination directions to reconstruct all Fourier components of an object in a spherical volume, a technique that is known as diffraction tomography [23

_{i}23. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. **1**(4), 153–156 (1969). [CrossRef]

24. M. G. Gustafsson, D. A. Agard, and J. W. Sedat, “Sevenfold improvement of axial resolution in 3D wide-field microscopy using two objective-lenses,” Proc. SPIE **2412**, 147–156 (1995). [CrossRef]

***denotes the complex conjugation and

*R*is the complex amplitude of the zero-order light (which we will later use as the reference wave), which is a plane wave travelling in the direction of wave vector

*k*.

_{0}*O*is the complex amplitude of the object wave (light scattered by the object). In Fourier space, Eq. (2) translates into:

*-k*as a result of the convolution with

_{0}*O*from the other terms in Eq. (2), this suggests that the three-dimensional image formation of

*R*O*performs the correct mapping of the object’s Fourier components within the accuracy of the first order Born approximation.

21. V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. **205**(2), 165–176 (2002). [CrossRef] [PubMed]

*k*and

_{1}*k*. The solid lines depict the shells containing the desired object information and the dashed lines represent the conjugated information.

_{2}25. T. Noda, S. Kawata, and S. Minami, “Three-dimensional phase contrast imaging by an annular illumination microscope,” Appl. Opt. **29**(26), 3810–3815 (1990). [CrossRef] [PubMed]

29. M. van ’t Hoff, V. de Sars, and M. Oheim, “A programmable light engine for quantitative single molecule TIRF and HILO imaging,” Opt. Express **16**(22), 18495–18504 (2008). [CrossRef] [PubMed]

## 3. Experimental set-up

## 4. Phase stepping interferometry

_{1}..I

_{4}, where the phase of the reference wave

## 5. Simulation of the amplitude transfer function

## 6. Experimental results

_{z}-axis. The recovered information lies within the region predicted by the simulated ATF, however towards the outside of the ATF, signal strength is weaker than predicted. By comparing Figs. 4(c) and 4(e), it is still evident that information within a finite volume can be transferred using rotating illumination, whereas the information in the pure axial illumination lies on a shell only.

## 7. Discussion

_{x}-k

_{z}plane. Using our approach, other illumination patterns could also be implemented: sweeping the laser spot on radial lines in the BFP would result in transfer functions that are symmetric to the k

_{x}-k

_{y}plane but anisotropic in the lateral plane. To increase the lateral symmetry, one could fuse data sets using radial line sweeps that were azimuthally rotated by 120° to synthesize a more symmetric transfer function.

1. F. Zernike, “How I discovered phase contrast,” Science **121**(3141), 345–349 (1955). [CrossRef] [PubMed]

30. G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett. **31**(6), 775–777 (2006). [CrossRef] [PubMed]

_{z}-direction is very limited, one cannot map the refractive index directly, instead a certain optical path length is measured. Furthermore the missing cone region introduces out-of-focus blur that can obscure the in-focus image.

17. W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods **4**(9), 717–719 (2007). [CrossRef] [PubMed]

16. F. Charrière, 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**(2), 178–180 (2006). [CrossRef] [PubMed]

31. S. Vertu, J.-J. Delaunay, I. Yamada, and O. Haeberlé, “Diffraction microtomography with sample rotation: influence of a missing apple core in the recorded frequency space,” Cent. Eur. J. Phys. **7**(1), 22–31 (2009). [CrossRef]

## 8. Outlook

**Appendix A: 3D image acquisition**

_{z}of 3D amplitude space. With now a new reference coordinate system fixed to the sample, each newly acquired 2D real space plane contributes another z-projection in Fourier space, but with an extra

**Appendix B: Derivation of the phase stepping algorithm**

*O*, we multiply the phase-shift angles on the images and sum the data sets up:

## References and links

1. | F. Zernike, “How I discovered phase contrast,” Science |

2. | R. D. Allen, G. B. David, and G. Nomarski, “The zeiss-Nomarski differential interference equipment for transmitted-light microscopy,” Z. Wiss. Mikrosk. |

3. | O. Shimomura, “The discovery of aequorin and green fluorescent protein,” J. Microsc. |

4. | L. Liu, J. R. Trimarchi, R. Oldenbourg, and D. L. Keefe, “Increased birefringence in the meiotic spindle provides a new marker for the onset of activation in living oocytes,” Biol. Reprod. |

5. | M. Shribak and S. Inoué, “Orientation-independent differential interference contrast microscopy,” Appl. Opt. |

6. | N. Lue, W. Choi, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, “Quantitative phase imaging of live cells using fast Fourier phase microscopy,” Appl. Opt. |

7. | G. Popescu, L. P. Deflores, J. C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, and M. S. Feld, “Fourier phase microscopy for investigation of biological structures and dynamics,” Opt. Lett. |

8. | E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holography for quantitative phase-contrast imaging,” Opt. Lett. |

9. | T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, “Hilbert phase microscopy for investigating fast dynamics in transparent systems,” Opt. Lett. |

10. | U. Schnars and W. P. Juptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. |

11. | F. Charrière, J. Kühn, T. Colomb, F. Montfort, E. Cuche, Y. Emery, K. Weible, P. Marquet, and C. Depeursinge, “Characterization of microlenses by digital holographic microscopy,” Appl. Opt. |

12. | H. Ding, F. Nguyen, S. A. Boppart, and G. Popescu, “Optical properties of tissues quantified by Fourier-transform light scattering,” Opt. Lett. |

13. | H. Ding, Z. Wang, F. Nguyen, S. A. Boppart, and G. Popescu, “Fourier transform light scattering of inhomogeneous and dynamic structures,” Phys. Rev. Lett. |

14. | B. Kemper, D. Carl, J. Schnekenburger, I. Bredebusch, M. Schafer, W. Domschke, and G. von Bally, “Investigation of living pancreas tumor cells by digital holographic microscopy,” J. Biomed. Opt. |

15. | S. S. Kou and C. J. Sheppard, “Imaging in digital holographic microscopy,” Opt. Express |

16. | F. Charrière, 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. |

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

18. | M. Debailleul, B. Simon, V. Georges, O. Haeberlé, and V. Lauer, “Holographic microscopy and diffractive microtomography of transparent samples,” Meas. Sci. Technol. |

19. | N. Fukutake and T. D. Milster, “Proposal of three-dimensional phase contrast holographic microscopy,” Opt. Express |

20. | Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express |

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

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

23. | E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. |

24. | M. G. Gustafsson, D. A. Agard, and J. W. Sedat, “Sevenfold improvement of axial resolution in 3D wide-field microscopy using two objective-lenses,” Proc. SPIE |

25. | T. Noda, S. Kawata, and S. Minami, “Three-dimensional phase contrast imaging by an annular illumination microscope,” Appl. Opt. |

26. | G. W. Ellis, “An Annular Scan Phase-Contrast Scanned Aperture Microscope (Aspsam),” Cell Motil. Cytoskeleton |

27. | A. L. Mattheyses and D. Axelrod, “Effective elimination of laser interference fringing in fluorescence microscopy by spinning azimuthal incidence angle,” Biophys. J. |

28. | R. Fiolka, Y. Belyaev, H. Ewers, and A. Stemmer, “Even illumination in total internal reflection fluorescence microscopy using laser light,” Microsc. Res. Tech. |

29. | M. van ’t Hoff, V. de Sars, and M. Oheim, “A programmable light engine for quantitative single molecule TIRF and HILO imaging,” Opt. Express |

30. | G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett. |

31. | S. Vertu, J.-J. Delaunay, I. Yamada, and O. Haeberlé, “Diffraction microtomography with sample rotation: influence of a missing apple core in the recorded frequency space,” Cent. Eur. J. Phys. |

**OCIS Codes**

(100.2000) Image processing : Digital image processing

(100.5070) Image processing : Phase retrieval

(100.6950) Image processing : Tomographic image processing

(110.0180) Imaging systems : Microscopy

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(100.3175) Image processing : Interferometric imaging

(110.6955) Imaging systems : Tomographic imaging

**ToC Category:**

Microscopy

**History**

Original Manuscript: May 11, 2009

Revised Manuscript: June 12, 2009

Manuscript Accepted: June 22, 2009

Published: July 20, 2009

**Virtual Issues**

Vol. 4, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Reto Fiolka, Kai Wicker, Rainer Heintzmann, and Andreas Stemmer, "Simplified approach to diffraction tomography in optical microscopy," Opt. Express **17**, 12407-12417 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-12407

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### References

- F. Zernike, “How I discovered phase contrast,” Science 121(3141), 345–349 (1955). [CrossRef] [PubMed]
- R. D. Allen, G. B. David, and G. Nomarski, “The zeiss-Nomarski differential interference equipment for transmitted-light microscopy,” Z. Wiss. Mikrosk. 69(4), 193–221 (1969). [PubMed]
- O. Shimomura, “The discovery of aequorin and green fluorescent protein,” J. Microsc. 217(1), 3–15 (2005). [CrossRef]
- L. Liu, J. R. Trimarchi, R. Oldenbourg, and D. L. Keefe, “Increased birefringence in the meiotic spindle provides a new marker for the onset of activation in living oocytes,” Biol. Reprod. 63(1), 251–258 (2000). [CrossRef] [PubMed]
- M. Shribak and S. Inoué, “Orientation-independent differential interference contrast microscopy,” Appl. Opt. 45(3), 460–469 (2006). [CrossRef] [PubMed]
- N. Lue, W. Choi, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, “Quantitative phase imaging of live cells using fast Fourier phase microscopy,” Appl. Opt. 46(10), 1836–1842 (2007). [CrossRef] [PubMed]
- G. Popescu, L. P. Deflores, J. C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, and M. S. Feld, “Fourier phase microscopy for investigation of biological structures and dynamics,” Opt. Lett. 29(21), 2503–2505 (2004). [CrossRef] [PubMed]
- E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holography for quantitative phase-contrast imaging,” Opt. Lett. 24(5), 291–293 (1999). [CrossRef]
- T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, “Hilbert phase microscopy for investigating fast dynamics in transparent systems,” Opt. Lett. 30(10), 1165–1167 (2005). [CrossRef] [PubMed]
- U. Schnars and W. P. Juptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33(2), 179–181 (1994). [CrossRef] [PubMed]
- F. Charrière, J. Kühn, T. Colomb, F. Montfort, E. Cuche, Y. Emery, K. Weible, P. Marquet, and C. Depeursinge, “Characterization of microlenses by digital holographic microscopy,” Appl. Opt. 45(5), 829–835 (2006). [CrossRef] [PubMed]
- H. Ding, F. Nguyen, S. A. Boppart, and G. Popescu, “Optical properties of tissues quantified by Fourier-transform light scattering,” Opt. Lett. 34(9), 1372–1374 (2009). [CrossRef] [PubMed]
- H. Ding, Z. Wang, F. Nguyen, S. A. Boppart, and G. Popescu, “Fourier transform light scattering of inhomogeneous and dynamic structures,” Phys. Rev. Lett. 101(23), 238102 (2008). [CrossRef] [PubMed]
- B. Kemper, D. Carl, J. Schnekenburger, I. Bredebusch, M. Schafer, W. Domschke, and G. von Bally, “Investigation of living pancreas tumor cells by digital holographic microscopy,” J. Biomed. Opt. 11(3), 034005–034008 (2006). [CrossRef]
- S. S. Kou and C. J. Sheppard, “Imaging in digital holographic microscopy,” Opt. Express 15(21), 13640–13648 (2007). [CrossRef] [PubMed]
- F. Charrière, 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(2), 178–180 (2006). [CrossRef] [PubMed]
- W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4(9), 717–719 (2007). [CrossRef] [PubMed]
- M. Debailleul, B. Simon, V. Georges, O. Haeberlé, and V. Lauer, “Holographic microscopy and diffractive microtomography of transparent samples,” Meas. Sci. Technol. 19(7), 074009 (2008). [CrossRef]
- N. Fukutake and T. D. Milster, “Proposal of three-dimensional phase contrast holographic microscopy,” Opt. Express 15(20), 12662–12679 (2007). [CrossRef] [PubMed]
- Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17(1), 266–277 (2009). [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(2), 165–176 (2002). [CrossRef] [PubMed]
- M. Born, and E. Wolf, Principles of optics,7th ed. (Cambridge University press, 2005).
- E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1(4), 153–156 (1969). [CrossRef]
- M. G. Gustafsson, D. A. Agard, and J. W. Sedat, “Sevenfold improvement of axial resolution in 3D wide-field microscopy using two objective-lenses,” Proc. SPIE 2412, 147–156 (1995). [CrossRef]
- T. Noda, S. Kawata, and S. Minami, “Three-dimensional phase contrast imaging by an annular illumination microscope,” Appl. Opt. 29(26), 3810–3815 (1990). [CrossRef] [PubMed]
- G. W. Ellis, “An Annular Scan Phase-Contrast Scanned Aperture Microscope (Aspsam),” Cell Motil. Cytoskeleton 10, 342–342 (1988).
- A. L. Mattheyses and D. Axelrod, “Effective elimination of laser interference fringing in fluorescence microscopy by spinning azimuthal incidence angle,” Biophys. J. 88, 341A–342A (2005).
- R. Fiolka, Y. Belyaev, H. Ewers, and A. Stemmer, “Even illumination in total internal reflection fluorescence microscopy using laser light,” Microsc. Res. Tech. 71(1), 45–50 (2008). [CrossRef]
- M. van ’t Hoff, V. de Sars, and M. Oheim, “A programmable light engine for quantitative single molecule TIRF and HILO imaging,” Opt. Express 16(22), 18495–18504 (2008). [CrossRef] [PubMed]
- G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett. 31(6), 775–777 (2006). [CrossRef] [PubMed]
- S. Vertu, J.-J. Delaunay, I. Yamada, and O. Haeberlé, “Diffraction microtomography with sample rotation: influence of a missing apple core in the recorded frequency space,” Cent. Eur. J. Phys. 7(1), 22-31 (2009). [CrossRef]

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