## Reconstruction of refractive-index distribution in off-axis digital holography optical diffraction tomographic system

Optics Express, Vol. 17, Issue 16, pp. 13758-13767 (2009)

http://dx.doi.org/10.1364/OE.17.013758

Acrobat PDF (39121 KB)

### Abstract

In the paper the optical diffraction tomographic system for reconstruction of the internal refractive index distribution in optical fiber utilizing grating Mach-Zehnder interferometer configuration is explored. The setup applies afocal imaging. Conventional grating application gives, however, highly aberrated object beam producing incorrect refractive-index reconstructions. The grating inherent aberrations are characterized, its influence on both image projections and refractive index reconstructions is presented. To remove aberrations and enable tomographic reconstruction a novel digital holographic algorithm, correcting optical system imaging, is developed. The algorithm uses plane wave spectrum decomposition of optical field for solving diffraction problem between parallel and tilted planes and enabling correction of imaging system aberrations. The algorithm concept was successfully proved in simulations and the experiment.

© 2009 Optical Society of America

## 1. Introduction

5. W. Górski and M. Kujawińska, “Three-dimensional reconstruction of refractive index inhomogeneities in optical phase elements,” Opt. Lasers Eng. **38**, 373–385 (2002). [CrossRef]

6. E.N. Leith and L. Shentu, “Tomographic reconstruction of objects by grating interferometer,” Appl. Opt. **25**, 907–913 (1986). [CrossRef] [PubMed]

7. L. Sałbut and M. Kujawińska, “The optical measurement station for complex testing of microelements,” Opt. Laser Eng. **36**, 225–240 (2001). [CrossRef]

8. R. Krajewski, M. Kujawińska, B. Volckaerts, and H. Thienpont “Low-cost microinterferomtric tomograpy system for 3D refraction index distribution Measurements in optical fiber splices, Proc. SPIE **5855**, 17th OFS conference Brugge, 347–351 (2005). [CrossRef]

*ϕ*the complex optical field scattered by the object is calculated using the temporal phase shifting method [10

10. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. , **22**, 1268–1270 (1997). [CrossRef] [PubMed]

11. A.J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrasonic imaging **4**, 336–350 (1982). [CrossRef] [PubMed]

13. T.C. Wedberg, J.J. Stamnes, and W. Singer, “Comparison of filtered backpropagation and backprojection algorithms for quantitative tomography,” Appl. Opt. **34**, 6575–6581 (1995). [CrossRef] [PubMed]

14. T. Kozacki, M. Kujawinska, and P. Kniazewski, “Investigating the limitation of optical scalar field tomography,” Opto-Electron. Rev. **15**, 102–109 (2007). [CrossRef]

15. P. Ferraro, S. De Nicola, A. Finizio, and G. Pierattini, “Reflective grating interferometer in a noncollimated configuration,” Appl. Opt. **39**, 2116–2121 (2000). [CrossRef]

## 2. Analysis of imaging system errors

^{th}and 0

^{th}respectively). Therefore one is unable to conjugate the object and the detector planes optically. The optical system conjugates detector and distorted image planes introducing severe aberrations.

*f*from object plane into a single image harmonics in the distorted image plane

_{xi}16. F. Shen and A. Wang, “Fast-Fourier transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. **45**, 1102–1110 (2006). [CrossRef] [PubMed]

17. K.-H. Brenner and W. Singer, “Light propagation through microlenses: a new simulation method,” Appl. Opt. **32**, 4984–4988 (1993). [CrossRef] [PubMed]

18. N. Delen and B. Hooker, “Free-space beam propagation between arbitrarily oriented planes based on full diffraction theory: a fast Fourier transform approach,” J. Opt. Soc. Am. A **15**, 857–867 (1998). [CrossRef]

19. K. Matsushima, H. Schimmel, and F. Wyrowski, “Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves,” J. Opt. Soc. Am. A **20**, 1755–1762 (2003). [CrossRef]

## 3 Digital holographic reconstruction of the projection image

### 3.1. Algorithm Concept

21. D. Lebrun, A. Benkouider, S. Coëtmellec, and M. Malek, “Particle field digital holographic reconstruction in arbitrary tilted planes,” Opt. Express **11**, 224–229 (2003). [CrossRef] [PubMed]

22. L. Yu, Y. An, and L. Cai, “Numerical reconstruction of digital holograms with variable viewing angles,” Opt. Express **10**, 1250–1257 (2002). [PubMed]

_{h}’ is propagated to the grating plane. At the grating plane X

_{g}the direction of propagation is changed into highly off-axis one. Finally such a field is propagated to an object centre (plane X

_{O}). The digital holographic algorithm could be composed of standard free space propagation algorithms and could perform the above mentioned job, unfortunately in an highly inefficient manner. The sampling have to be very dense and the size of processed data will be very large (both on-axis and off-axis fields are processed by the algorithm). Additionally splitting algorithm into unnecessary steps highly enlarges algorithm computational load. Therefore we have elaborated a special holographic reconstruction algorithm.

**x**=

*xê*

*+*

_{x}*yê*

*,*

_{y}**f**

*=*

_{t}*f*

_{x}ê*+*

_{x}*f*

_{y}ê*. The*

_{y}*ê*

*are the unit vectors and dot denotes the vector scalar product. The spatial frequencies ft are connected with wave vector*

_{x,y,z}**k**=2

*π*(

*ê*

*+*

_{x}f_{x}*ê*

*+*

_{y}f_{y}*ê*

*(*

_{z}*λ*

^{-2}-

**f**·

_{t}**f**)

_{t}^{1/2}). The evanescent waves are not considered within this paper. The above formula allows to decompose optical field into a collection of plane waves characterized by wave vectors and complex amplitudes. Such plane waves can be defined in an arbitrarily tilted coordinate system.

_{h}’ into a plane wave spectrum decomposition defined in the tilted object projection image plane X

_{O}’ using propagation algorithm between parallel planes; step 2 - conversion of the plane wave spectrum decomposition defined in the tilted object projection image plane X

_{O}’ into optical field in the object projection image plane X

_{O}using propagation algorithm between two tilted planes.

### 3.2. Algorithm step 1 - propagation between parallel planes

_{2}=5 mm, the required sampling Δ

_{x}≈0.3 µm, the spatial shift 3.14 mm, we need

*N*=13000 sampling points.

_{x}**f**

_{m}=[-

*f*,

_{m}*f*]. In diffraction tomography slow variations of refractive index distribution are characterized only. Additionally, during the tomographic reconstruction the low pass filtering operations are applied. For computational simplicity it is convenient to design algorithm (Eq. (3)) working with the shifted frequency

_{m}**f**

_{ts}=(

*f*=

_{xs}*f*-Λ

_{x}^{-1},

*f*):

_{y}_{O’}:

24. T. Kozacki, “Numerical errors of diffraction computing using plane wave spectrum decomposition,” Opt. Commun. **281**, 4219–4223 (2008). [CrossRef]

*x*≤

_{lf}*N*Δ

_{x}*/2. This gives required number of samples:*

_{x}*λ*=0.532 µm, Λ=1 µm,

*z*=6.459 mm,

_{1}*z*=-5 mm and

_{2}*f*=326.4 mm

_{m}^{-1}(such a bandwidth corresponds 10° aperture angle), we get necessary computational window dimension N

_{x}>1594. By using computation window of that size, we ensure that the aliasing in the phase accommodation kernel occurs at higher frequency than fm and the entire optical field bandwidth is processed correctly.

### 3.3. Algorithm step 2 - propagation between tilted planes

**k**′=

*k*′

_{x}ê*+*

_{x}*k*′

*+*

_{y}ê*k*′

_{z}ê*defined in plane (x’, y, z’) is transferred into*

_{z}**k**=

**Tk**′ defined in (x, y, z), where

**T**, the full collection of the plane waves defining an arbitrary optical field can be characterized in a new rotated coordinate system. In this way we can perform mapping of plane wave spectra between planes X

_{O’}and X

_{O}by

25. T.M. Lehmann, C. Gonner, and K. Spitzer, “Survey: interpolation methods in medical image processing,” IEEE Trans. Med. Imaging **18**, 1049–1075 (1999). [CrossRef]

*U*′(

^{X}**f**

*) by the carrier Λ*

_{t}^{-1}. Such a frequency shift unnecessarily enlarges computer power consumption of algorithm and requires dense sampling. For this reason we propose application of a modified mapping with shifted frequency coordinates:

^{-1}can be applied.

_{O’}and projection image plane X

_{O}by formula:

### 3.4. Experimental verification of the algorithm

## 4. Conclusions

## Acknowledgements

## References and links

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

2. | W. L. Howes and D. R. Buchele, “Optical Interferometry of Inhomogeneous Gasses,” J. Opt. Soc. Am. |

3. | L. P. Yaroslavskii and N. Merzlyakov, |

4. | S. K. Mangal and K. Ramesh, “Determination of characteristic parameters in integrated photoelasticity by phase-shifting technique,” Opt. Laser Eng. |

5. | W. Górski and M. Kujawińska, “Three-dimensional reconstruction of refractive index inhomogeneities in optical phase elements,” Opt. Lasers Eng. |

6. | E.N. Leith and L. Shentu, “Tomographic reconstruction of objects by grating interferometer,” Appl. Opt. |

7. | L. Sałbut and M. Kujawińska, “The optical measurement station for complex testing of microelements,” Opt. Laser Eng. |

8. | R. Krajewski, M. Kujawińska, B. Volckaerts, and H. Thienpont “Low-cost microinterferomtric tomograpy system for 3D refraction index distribution Measurements in optical fiber splices, Proc. SPIE |

9. | R. Czarnek, “High sensitivity moiré interferometry with compact achromatic interferometer,” Opt. Laser Eng. |

10. | I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. , |

11. | A.J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrasonic imaging |

12. | J. Hseigh, |

13. | T.C. Wedberg, J.J. Stamnes, and W. Singer, “Comparison of filtered backpropagation and backprojection algorithms for quantitative tomography,” Appl. Opt. |

14. | T. Kozacki, M. Kujawinska, and P. Kniazewski, “Investigating the limitation of optical scalar field tomography,” Opto-Electron. Rev. |

15. | P. Ferraro, S. De Nicola, A. Finizio, and G. Pierattini, “Reflective grating interferometer in a noncollimated configuration,” Appl. Opt. |

16. | F. Shen and A. Wang, “Fast-Fourier transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. |

17. | K.-H. Brenner and W. Singer, “Light propagation through microlenses: a new simulation method,” Appl. Opt. |

18. | N. Delen and B. Hooker, “Free-space beam propagation between arbitrarily oriented planes based on full diffraction theory: a fast Fourier transform approach,” J. Opt. Soc. Am. A |

19. | K. Matsushima, H. Schimmel, and F. Wyrowski, “Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves,” J. Opt. Soc. Am. A |

20. | C. J. Dasch, “One-dimensional tomography: a comparison of Abel, onion-peeling, and filtered backprojection methods,” Appl. Opt. |

21. | D. Lebrun, A. Benkouider, S. Coëtmellec, and M. Malek, “Particle field digital holographic reconstruction in arbitrary tilted planes,” Opt. Express |

22. | L. Yu, Y. An, and L. Cai, “Numerical reconstruction of digital holograms with variable viewing angles,” Opt. Express |

23. | J. J. Stamnes, |

24. | T. Kozacki, “Numerical errors of diffraction computing using plane wave spectrum decomposition,” Opt. Commun. |

25. | T.M. Lehmann, C. Gonner, and K. Spitzer, “Survey: interpolation methods in medical image processing,” IEEE Trans. Med. Imaging |

**OCIS Codes**

(090.1000) Holography : Aberration compensation

(110.6960) Imaging systems : Tomography

(090.1995) Holography : Digital holography

**ToC Category:**

Holography

**History**

Original Manuscript: March 25, 2009

Revised Manuscript: May 18, 2009

Manuscript Accepted: June 18, 2009

Published: July 24, 2009

**Citation**

Tomasz Kozacki, Rafal Krajewski, and Małgorzata Kujawińska, "Reconstruction of refractive-index distribution in off-axis digital holography optical diffraction tomographic system," Opt. Express **17**, 13758-13767 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-13758

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### References

- M. Born and E. Wolf, Principles of Optics 7th (expanded) edition, (Cambridge University Press, 1999).
- W. L. Howes and D. R. Buchele, "Optical Interferometry of Inhomogeneous Gasses," J. Opt. Soc. Am. 56, 1517-1528 (1966). [CrossRef]
- L. P. Yaroslavskii and N. Merzlyakov, Methods of digital holography (Consultants Bureau, New York, 1980).
- S. K. Mangal, and K. Ramesh, "Determination of characteristic parameters in integrated photoelasticity by phase-shifting technique," Opt. Laser Eng. 31, 263-278 (1999). [CrossRef]
- W. Górski and M. Kujawińska, "Three-dimensional reconstruction of refractive index inhomogeneities in optical phase elements," Opt. Lasers Eng. 38,373-385 (2002). [CrossRef]
- E.N. Leith and L. Shentu, "Tomographic reconstruction of objects by grating interferometer," Appl. Opt. 25, 907-913 (1986). [CrossRef] [PubMed]
- L. Sałbut and M. Kujawińska, "The optical measurement station for complex testing of microelements," Opt. Laser Eng. 36, 225-240 (2001) . [CrossRef]
- R. Krajewski, M. Kujawińska, B. Volckaerts, and H. Thienpont "Low-cost microinterferomtric tomograpy system for 3D refraction index distribution Measurements in optical fiber splices, Proc. SPIE 5855, 17th OFS conference Brugge, 347-351 (2005). [CrossRef]
- R. Czarnek, "High sensitivity moiré interferometry with compact achromatic interferometer," Opt. Laser Eng. 13, 93-101 (1990).
- I. Yamaguchi and T. Zhang, "Phase-shifting digital holography," Opt. Lett. 22, 1268-1270 (1997). [CrossRef] [PubMed]
- A.J. Devaney, "A filtered backpropagation algorithm for diffraction tomography," Ultrasonic imaging 4, 336-350 (1982). [CrossRef] [PubMed]
- J. Hseigh, Computed Tomograph, (SPIE Press, Washington, 2003).
- T.C. Wedberg, J.J. Stamnes and W. Singer, "Comparison of filtered backpropagation and backprojection algorithms for quantitative tomography," Appl. Opt. 34, 6575-6581 (1995). [CrossRef] [PubMed]
- T. Kozacki, M. Kujawinska and P. Kniazewski, "Investigating the limitation of optical scalar field tomography," Opto-Electron.Rev. 15, 102-109 (2007). [CrossRef]
- P. Ferraro, S. De Nicola, A. Finizio, and G. Pierattini, "Reflective grating interferometer in a noncollimated configuration," Appl. Opt. 39, 2116-2121 (2000). [CrossRef]
- F. Shen and A. Wang, "Fast-Fourier transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula," Appl. Opt. 45, 1102-1110 (2006). [CrossRef] [PubMed]
- K.-H. Brenner, and W. Singer, "Light propagation through microlenses: a new simulation method," Appl. Opt. 32, 4984-4988 (1993). [CrossRef] [PubMed]
- Q4. N. Delen and B. Hooker, "Free-space beam propagation between arbitrarily oriented planes based on full diffraction theory: a fast Fourier transform approach," J. Opt. Soc. Am. A 15, 857-867 (1998). [CrossRef]
- Q5. K. Matsushima, H. Schimmel, and F. Wyrowski, "Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves," J. Opt. Soc. Am. A 20, 1755-1762 (2003). [CrossRef]
- C. J. Dasch, "One-dimensional tomography: a comparison of Abel, onion-peeling, and filtered backprojection methods," Appl. Opt. 31, 1146-1152 (1992). [CrossRef] [PubMed]
- D. Lebrun, A. Benkouider, S. Coëtmellec, and M. Malek, "Particle field digital holographic reconstruction in arbitrary tilted planes," Opt. Express 11, 224-229 (2003). [CrossRef] [PubMed]
- L. Yu, Y. An, and L. Cai, "Numerical reconstruction of digital holograms with variable viewing angles," Opt. Express 10, 1250-1257 (2002). [PubMed]
- J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol 1986).
- T. Kozacki, "Numerical errors of diffraction computing using plane wave spectrum decomposition," Opt. Commun. 281, 4219-4223 (2008). [CrossRef]
- T.M. Lehmann, C. Gonner, and K. Spitzer, "Survey: interpolation methods in medical image processing," IEEE Trans. Med. Imaging 18, 1049-1075 (1999). [CrossRef]

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