## Reconstruction of a complex object from two in-line holograms

Optics Express, Vol. 11, Issue 6, pp. 572-578 (2003)

http://dx.doi.org/10.1364/OE.11.000572

Acrobat PDF (437 KB)

### Abstract

A novel numerical algorithm is proposed to reconstruct a complex object from two Gabor in-line holograms. With this algorithm, both the real and imaginary parts of the complex amplitude of the wave front in the object field can be retrieved, and the “twin-image” noise is eliminated at the same time. Therefore, the complex refractive index of the object can be obtained without disturbance. Digital simulations are given to prove the effectiveness of this algorithm. Some practical experimental conditions are investigated by use of error estimation.

© 2003 Optical Society of America

## 1. Introduction

1. D. Sayer, J. Kirz, R. Feder, D. M. Kim, and E. Spiller, “Potential operating region for ultrasoft X-ray microscopy of biological materials,” Science **196**, 1339–1340 (1977). [CrossRef]

3. I. McNulty, “The future of X-ray holography,” Nucl. Instrum. Methods A **347**, 170–176 (1994). [CrossRef]

4. C. Jacobsen, M. R. Howells, J. Kirz, and S. Rothman, “X-ray holographic microscopy using photoresists,” J. Opt. Soc. Am. A **7**, 1847–1861 (1990). [CrossRef]

5. P. Spanne, C. Raven, I. Snigireva, and A. Snigirev, “In-line holography and phase-contrast microtomography with high energy x-rays,” Phys. Med. Biol. **44**(3), 741–749 (1999). [CrossRef] [PubMed]

7. K. Hirano and A. Momose, “Development of an X-ray interferometer for high-resolution phase-contrast X-ray imaging,” Jpn. J. Appl. Phys. **38**, L1556–L1558 (1999). [CrossRef]

8. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. **21**, 2758–2769 (1982). [CrossRef] [PubMed]

9. G. Liu and P. D. Scott, “Phase retrieval and twin-image elimination for in-line Fresnel holograms,” J. Opt. Soc. Am. A **4**, 159–165 (1987). [CrossRef]

11. M. H. Maleki and A. J. Devaney, “Noniterative reconstruction of complex-valued objects from two intensity measurements,” Opt. Eng. **33**, 3243–3253 (1994). [CrossRef]

*et al.*proposed another algorithm to eliminate the twin-image noise by subtracting a hologram recorded in the double distance from the reconstruction of the hologram recorded in the single distance [12

12. T. Xiao, H. Xu, Y. Zhang, J. Chen, and Z. Xu, “Digital image decoding for in-line X-ray holography using two holograms,” J. Mod. Opt. **45**, 343–353 (1998). [CrossRef]

13. L. Xu, J. Miao, and A. Asundi, “Properties of digital holography based on in-line configuration,” Opt. Eng. **39**, 3214–3219 (2000). [CrossRef]

## 2. Theory

*X,Y*) has a transmittance of

*t*(

*x,y*) and phase shift ϕ(

*x,y*), the complex amplitude of the object wave front Ũ(

*x,y*) becomes

_{0}(

*x,y*) is the incident wave front and

*a*(

*x,y*) is the pure absorption of the object. Let us assume that Ũ

_{0}(

*x,y*)=1, which indicates a plane-wave incident condition, and assume that

*a*(

*x,y*)≪1, | φ(

*x,y*)|≪1. These assumptions can be easily satisfied because the complex refractive indices of all elements in high-energy x-ray regions are always near to 1. If the size of the object is not too large (less than 10 mm) and the object does not contain too many heavy atoms, the absorption and phase shift will be quite small. Then we have

*a*(

*x,y*)ϕ(

*x,y*) in Eq. (2) and in the following equations. Here we define a complex absorption factor

*ã*(

*x,y*)=

*a*(

*x,y*)-

*i*φ(

*x,y*). Thus Re[

*ã*(

*x,y*)]=

*a*(

*x,y*) is the pure absorption and -Im[

*ã*(

*x,y*)]=φ(

*x,y*) is the phase shift.

*z*is the distance between the object and the hologram. The exponential term exp[

*i2*π

*z*/λ] is a constant in a given (

*x,y*) plane, and it will only add a constant phase shift to each pixel; thus we will not take this term into account in the following discussion. Now we define a convolution kernel function

*h*(

_{z}*x,y*) as

*z*and

*2z*, we will have two intensity distributions

*I*(

_{1}*x,y*) and

*I*(

_{2}*x,y*):

*h*(

_{a}*x,y*) *

*h*(

_{b}*x,y*)=

*h*(

_{a+b}*x,y*),

*h**

*(*

_{z}*x,y*)=-

*h*(

_{z}*x,y*),

*h*

_{0}(

*x,y*)=δ(

*x,y*) [10], we can obtain the following result,

*ã*(

*x,y*) as

*I*′(

*x,y*) as

*I*(

_{1}*x,y*) and

*a*(

_{r}*x,y*) are all known to us, then

*I*′(

*x,y*) becomes known as well. Then we can obtain

*a*(

_{i}*x,y*) from

*I*′(

*x,y*). In fact, from Eq. (11) we can obtain the following relations:

*2nz*. If this distance is large enough, the effect from errors is small and can be neglected. Therefore, we can obtain the distribution of

*a*as

_{i}## 3. Simulations

*z*=1 mm; discrete sampling number

*N*, 256; and total view field width, 16 µm. The absorption of the object is 10%, and the phase shift varies from 0 to 0.07, which is shown in Fig. 1 as curves (a) and (d). The reconstructions using our algorithm and an optical algorithm are shown in Fig. 1, from which we can see that the absorption and the phase shift are clearly retrieved by our algorithm.

*M*in Eq. (14) and the position error of

*z*in Eq. (3). In addition, because our algorithm is based on the weak absorption and weak phase-shift assumption according to Eq. (2), we study the reconstruction’s error as a function of the magnitude of the absorption and phase shift as well. Thus we define the rms absorption and phase errors as

*M*will affect the result of phase shift

*E*only. The relations of

_{p}*M-E*with different

_{p}*N*are shown in Fig. 2(a). From these curves in Fig. 2(a) we can find that the minimum

*E*appear at different

_{p}*M*values for different

*N*; the optimized value of

*M*is approximately 17 or 45 for

*N*equal to 256 or 512. For large

*N*value, optimized

*M*value also becomes larger. Figures 2(b) and 2(c) are the

*E*and

_{a}*E*with different absorption and phase-shift magnitudes, from which we can find that the errors increase greatly when the magnitude of absorption or phase shift increases. Thus this algorithm is valid only for weak absorption and weak phase-shift situations, as we assumed before in Eq. (2).

_{p}*z*and lateral displacement error Δ

*x*. The effect of these errors can be studied by digital simulations. First, we keep the recording distance

*z*of the first hologram for 1 mm and vary the recording distance of the second hologram to see the effect caused by distance error. The distance of 2

*z*is varied from 1.7 to 2.3 mm. The reconstructions are calculated as the function of 2

*z*, shown in Fig. 3. The rms errors of the reconstruction are shown in Fig. 4(a). If we want both the rms absorption error and phase error to be less than 1×10

^{-4}, the distance of the second hologram should be in the range of 1.92–2.08 mm, which means a position accuracy of ±4%. Figure 4(b) shows the rms errors as the function of lateral displacement Δ

*x*, from which we can find that the lateral displacement has a magnificent effect on the reconstructions. The spacing interval of the plotted data in Fig. 4(b) is 1 pixel. Therefore, a lateral position accuracy of ±3 pixels or ±0.2 µm is need to ensure that the rms error is less than 1×10

^{-3}.

*M*=45. Not only the cell-like object but also the two squares are clearly reconstructed in this figure.

## 4. Conclusion

## Acknowledgment

## References and links

1. | D. Sayer, J. Kirz, R. Feder, D. M. Kim, and E. Spiller, “Potential operating region for ultrasoft X-ray microscopy of biological materials,” Science |

2. | M. R. Howells and C. J. Jacobsen, “X-ray holography,” Synchrotron Radiation News , |

3. | I. McNulty, “The future of X-ray holography,” Nucl. Instrum. Methods A |

4. | C. Jacobsen, M. R. Howells, J. Kirz, and S. Rothman, “X-ray holographic microscopy using photoresists,” J. Opt. Soc. Am. A |

5. | P. Spanne, C. Raven, I. Snigireva, and A. Snigirev, “In-line holography and phase-contrast microtomography with high energy x-rays,” Phys. Med. Biol. |

6. | T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard X-ray quantitative noninterferometric phase-contrast microscopy,” J. Phys. D |

7. | K. Hirano and A. Momose, “Development of an X-ray interferometer for high-resolution phase-contrast X-ray imaging,” Jpn. J. Appl. Phys. |

8. | J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. |

9. | G. Liu and P. D. Scott, “Phase retrieval and twin-image elimination for in-line Fresnel holograms,” J. Opt. Soc. Am. A |

10. | L. Onural and P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. |

11. | M. H. Maleki and A. J. Devaney, “Noniterative reconstruction of complex-valued objects from two intensity measurements,” Opt. Eng. |

12. | T. Xiao, H. Xu, Y. Zhang, J. Chen, and Z. Xu, “Digital image decoding for in-line X-ray holography using two holograms,” J. Mod. Opt. |

13. | L. Xu, J. Miao, and A. Asundi, “Properties of digital holography based on in-line configuration,” Opt. Eng. |

**OCIS Codes**

(090.0090) Holography : Holography

(100.3010) Image processing : Image reconstruction techniques

(100.5070) Image processing : Phase retrieval

(340.7460) X-ray optics : X-ray microscopy

**ToC Category:**

Research Papers

**History**

Original Manuscript: January 17, 2003

Revised Manuscript: March 14, 2003

Published: March 24, 2003

**Citation**

Yuxuan Zhang and Xinyi Zhang, "Reconstruction of a complex object from two in-line holograms," Opt. Express **11**, 572-578 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-6-572

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### References

- D. Sayer, J. Kirz, R. Feder, D. M. Kim, and E. Spiller, �??Potential operating region for ultrasoft X-ray microscopy of biological materials,�?? Science 196, 1339-1340 (1977). [CrossRef]
- M. R. Howells and C. J. Jacobsen, �??X-ray holography,�?? Synchrotron Radiation News, 3(4), 23-28 (1990). [CrossRef]
- I. McNulty, �??The future of X-ray holography,�?? Nucl. Instrum. Methods A 347, 170-176 (1994). [CrossRef]
- C. Jacobsen, M. R. Howells, J. Kirz, and S. Rothman, �??X-ray holographic microscopy using photoresists,�?? J. Opt. Soc. Am. A 7, 1847-1861 (1990). [CrossRef]
- P. Spanne, C. Raven, I. Snigireva, and A. Snigirev, �??In-line holography and phase-contrast microtomography with high energy x-rays,�?? Phys. Med. Biol. 44(3), 741-749 (1999). [CrossRef] [PubMed]
- T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, �??Hard X-ray quantitative noninterferometric phase-contrast microscopy,�?? J. Phys. D 32, 563-567 (1999). [CrossRef]
- K. Hirano and A. Momose, �??Development of an X-ray interferometer for high-resolution phase-contrast X-ray imaging,�?? Jpn. J. Appl. Phys. 38, L1556-L1558 (1999). [CrossRef]
- J. R. Fienup, �??Phase retrieval algorithms: a comparison,�?? Appl. Opt. 21, 2758-2769 (1982). [CrossRef] [PubMed]
- G. Liu and P. D. Scott, �??Phase retrieval and twin-image elimination for in-line Fresnel holograms,�?? J. Opt. Soc. Am. A 4, 159-165 (1987). [CrossRef]
- L. Onural and P. D. Scott, �??Digital decoding of in-line holograms,�?? Opt. Eng. 26, 1124-1132 (1987).
- M. H. Maleki and A. J. Devaney, �??Noniterative reconstruction of complex-valued objects from two intensity measurements,�?? Opt. Eng. 33, 3243-3253 (1994). [CrossRef]
- T. Xiao, H. Xu, Y. Zhang, J. Chen, and Z. Xu, �??Digital image decoding for in-line X-ray holography using two holograms,�?? J. Mod. Opt. 45, 343-353 (1998). [CrossRef]
- L. Xu, J. Miao, and A. Asundi, �??Properties of digital holography based on in-line configuration,�?? Opt. Eng. 39, 3214�??3219 (2000). [CrossRef]

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