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

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References

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Appl. Opt. (1)

J. Mod. Opt. (1)

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]

J. Opt. Soc. Am. A (2)

J. Phys. D (1)

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]

Jpn. J. Appl. Phys. (1)

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]

Nucl. Instrum. Methods A (1)

I. McNulty, �??The future of X-ray holography,�?? Nucl. Instrum. Methods A 347, 170-176 (1994).
[CrossRef]

Opt. Eng. (3)

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]

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

Phys. Med. Biol. (1)

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]

Science (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]

Synchrotron Radiation News (1)

M. R. Howells and C. J. Jacobsen, �??X-ray holography,�?? Synchrotron Radiation News, 3(4), 23-28 (1990).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

One-dimensional simulation results. Absorption (upper) and phase-shift (bottom) profile curve of the object, (a) and (d) real object, (b) and (e) reconstruction by two-intensity algorithm, (c) and (f) reconstruction by optical algorithm.

Fig. 2.
Fig. 2.

rms error: (a) Ep as the function of M, (b) and (c) Ea and Ep as the function of phase shift with different absorption.

Fig. 3.
Fig. 3.

Reconstructions with different recording distance of the second hologram (N=256, M=17): (a) absorption, (b) phase shift.

Fig. 4.
Fig. 4.

rms errors with different positioning error (N=256, M=17): (a) with distance error of 2z, (b) with lateral alignment error Δx.

Fig. 5.
Fig. 5.

Digital simulations of a cell-like sample: (a) original object, (b) and (c) holograms in distance of z and 2z, (d) reconstruction of the real part, (e) reconstruction of the imaginary part (M=45, N=512).

Equations (23)

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Ũ ( x , y ) = Ũ 0 ( x , y ) t ( x , y ) e i ϕ ( x , y )
= Ũ 0 ( x , y ) [ 1 - a ( x , y ) ] e i ϕ ( x , y ) ,
Ũ ( x , y ) [ 1 a ( x , y ) ] [ 1 + i ϕ ( x , y ) ]
= 1 a ( x , y ) + i ϕ ( x , y ) i a ( x , y ) ϕ ( x , y )
1 ã ( x , y ) .
Ũ h ( x , y ) = 1 i λ z exp ( i 2 π z λ ) Ũ ( ξ , η ) exp { i π λz [ ( x ξ ) 2 + ( y η ) 2 ] } d ξ d η ,
h z ( x , y ) = i λz exp [ i π λz ( x 2 + y 2 ) ] ;
Ũ h ( x , y ) = Ũ ( x , y ) h z ( x , y ) ,
I h ( x , y ) = Ũ h ( x , y ) Ũ h ( x , y )
1 ã ( x , y ) h z ( x , y ) ã ( x , y ) h z ( x , y ) .
{ I 1 ( x , y ) = 1 ã ( x , y ) h z ( x , y ) ã ( x , y ) h z ( x , y ) I 2 ( x , y ) = 1 ã ( x , y ) h 2 z ( x , y ) ã ( x , y ) h 2 z ( x , y ) .
I 1 h z + I 1 h z I 2 = 1 ( ã + ã ) .
Re [ ã ] = ( I 2 + 1 ) 2 I 1 Re [ h z ] .
I 1 = 1 a r h z a r h z + i ( a i h z - a i h z ) .
I ' = i ( 1 I 1 a r h z a r h z )
= a i h z a i h z .
I ' h z = a i a i h 2 z ,
I ' h 3 z = a i h 2 z a i h 4 z ,
I ' h 5 z = a i h 4 z a i h 6 z ,
a i = I ' h z + I ' h 3 z + + I ' h ( 2 n 1 ) z + a i h 2 n z .
Im [ ã ] = a i = I ' k = 1 M h ( 2 k 1 ) z .
E a = x { Re [ ã ( x ) ] a ( x ) } 2 N ,
E p = x { Im [ ã ( x ) ] ϕ ( x ) } 2 N .

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