Abstract

A new approach to the numerical reconstruction of wave fronts stored by in-line holography is presented. The new algorithm can achieve good reconstructed results in both unitary and nonunitary systems. The influences of recording distance and noise as well as of digitalization errors on the quality of reconstruction are numerically investigated. The experimental results demonstrate the validity of this new approach.

© 2003 Optical Society of America

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References

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1999

S. Lai, B. Kemper, G. V. Belly, “Off-axis reconstructions of in-line holograms for twin-image elimination,” Opt. Commun. 169, 37–43 (1999).
[CrossRef]

1998

T. Xian, H. Xu, Y. Zhang, J. Chen, Z. Xu, “Digital image decoding for in-line x-ray holography using two holograms,” J. Mod. Opt. 45, 343–353 (1998).

Y. Zhang, B. Z. Dong, B. Y. Gu, G. Z. Yang, “Beam shaping in the fractional Fourier domain,” J. Opt. Soc. Am. A 15, 1114–1120 (1998).
[CrossRef]

1997

1996

1994

1990

1987

1984

P. H. Malyak, B. J. Thompson, “Particle displacement and velocity measurement using holography,” Opt. Eng. 23, 567–576 (1984).
[CrossRef]

1982

1972

R. W. Gerchberg, W. O. Saxton, “Practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

1948

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[CrossRef] [PubMed]

Belly, G. V.

S. Lai, B. Kemper, G. V. Belly, “Off-axis reconstructions of in-line holograms for twin-image elimination,” Opt. Commun. 169, 37–43 (1999).
[CrossRef]

Chen, J.

T. Xian, H. Xu, Y. Zhang, J. Chen, Z. Xu, “Digital image decoding for in-line x-ray holography using two holograms,” J. Mod. Opt. 45, 343–353 (1998).

Dong, B. Z.

Ersoy, O. K.

Fienup, J. R.

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[CrossRef] [PubMed]

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “Practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chap. 2.

Gu, B. Y.

Howells, M. R.

Jacobsen, C.

Kemper, B.

S. Lai, B. Kemper, G. V. Belly, “Off-axis reconstructions of in-line holograms for twin-image elimination,” Opt. Commun. 169, 37–43 (1999).
[CrossRef]

Kirz, J.

Kreyszig, E.

E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, New York, 1986).

Lai, S.

S. Lai, B. Kemper, G. V. Belly, “Off-axis reconstructions of in-line holograms for twin-image elimination,” Opt. Commun. 169, 37–43 (1999).
[CrossRef]

Liu, G.

Malyak, P. H.

P. H. Malyak, B. J. Thompson, “Particle displacement and velocity measurement using holography,” Opt. Eng. 23, 567–576 (1984).
[CrossRef]

Onural, L.

L. Onural, P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).
[CrossRef]

Rothman, S.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “Practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Scott, P. D.

Thompson, B. J.

P. H. Malyak, B. J. Thompson, “Particle displacement and velocity measurement using holography,” Opt. Eng. 23, 567–576 (1984).
[CrossRef]

Xian, T.

T. Xian, H. Xu, Y. Zhang, J. Chen, Z. Xu, “Digital image decoding for in-line x-ray holography using two holograms,” J. Mod. Opt. 45, 343–353 (1998).

Xu, H.

T. Xian, H. Xu, Y. Zhang, J. Chen, Z. Xu, “Digital image decoding for in-line x-ray holography using two holograms,” J. Mod. Opt. 45, 343–353 (1998).

Xu, Z.

T. Xian, H. Xu, Y. Zhang, J. Chen, Z. Xu, “Digital image decoding for in-line x-ray holography using two holograms,” J. Mod. Opt. 45, 343–353 (1998).

Yang, G. Z.

Zhang, G. Q.

Zhang, Y.

T. Xian, H. Xu, Y. Zhang, J. Chen, Z. Xu, “Digital image decoding for in-line x-ray holography using two holograms,” J. Mod. Opt. 45, 343–353 (1998).

Y. Zhang, B. Z. Dong, B. Y. Gu, G. Z. Yang, “Beam shaping in the fractional Fourier domain,” J. Opt. Soc. Am. A 15, 1114–1120 (1998).
[CrossRef]

Zhuang, J.

Appl. Opt.

J. Mod. Opt.

T. Xian, H. Xu, Y. Zhang, J. Chen, Z. Xu, “Digital image decoding for in-line x-ray holography using two holograms,” J. Mod. Opt. 45, 343–353 (1998).

J. Opt. Soc. Am. A

Nature

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[CrossRef] [PubMed]

Opt. Commun.

S. Lai, B. Kemper, G. V. Belly, “Off-axis reconstructions of in-line holograms for twin-image elimination,” Opt. Commun. 169, 37–43 (1999).
[CrossRef]

Opt. Eng.

P. H. Malyak, B. J. Thompson, “Particle displacement and velocity measurement using holography,” Opt. Eng. 23, 567–576 (1984).
[CrossRef]

L. Onural, P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).
[CrossRef]

Optik (Stuttgart)

R. W. Gerchberg, W. O. Saxton, “Practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Other

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chap. 2.

E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, New York, 1986).

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Figures (10)

Fig. 1
Fig. 1

Schematic of the setup for in-line hologram recording.

Fig. 2
Fig. 2

Variation of the ratio B/ C with recording distance d.

Fig. 3
Fig. 3

Simulated one-dimensional object.

Fig. 4
Fig. 4

Dependence of the SSE on recording distance.

Fig. 5
Fig. 5

Reconstructed images obtained (a) by the GS algorithm and (b) by the YG algorithm after 200 iterations in a unitary system.

Fig. 6
Fig. 6

Reconstructed image obtained (a) by the GS algorithm and (b) by the YG algorithm after 200 iterations in a nonunitary system.

Fig. 7
Fig. 7

Relationship between the SSE and the number of iterations.

Fig. 8
Fig. 8

Description of the SSE as a function of reconstruction distance for (solid curve) the GS algorithm and (dashed curve) the YG algorithm.

Fig. 9
Fig. 9

Reconstruction of a real digital hologram recorded at d = 8.64 × 104 μm: (a) digitalized hologram, (b) result obtained by the GS algorithm after five iterations, (c) result obtained by the YG algorithm after five iterations.

Fig. 10
Fig. 10

Reconstruction of a real digital hologram recorded at d = 9.50 × 104 μm: (a) digitalized hologram, (b) result obtained by the GS algorithm after five iterations, (c) result obtained by the YG algorithm after five iterations.

Equations (17)

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U2xx, y2=Biλdexpi2πdλ  U1x1, y1×expiπλdx1-x22+y1-y22dx1dy1,
I2x2, y2=|U2x2, y2|2.
U2x2=GU1x1,
G*G=AI.
U1l=ρ1l expiφ1l, l=1, 2, 3,, N1,
U2m=ρ2m expiφ2m,
U2m=l=1N1GmlU1l, m=1, 2, 3,, N2.
Dρ1, φ1, ρ2, φ2=U2-GU1=m=1N2 |U2m-GU1m|21/2.
Δφ2D2=0, Δρ1D2=0,
φ2k=Argj=1N1Gkjρ1j expiφ1j,
ρ1k=1AkkAbsj=1N2Gjk*ρ2j expiϕ2j-jkAkjρ1j expiϕ1j,
φ2=ArgGU1,
ρ1=AbsG*U2.
SSE= ρ2-Gρ1n expiφ12/ ρ22,
C=1N1i=1N1|Aii|,
B=1N1N1-1i=1N1jiN1|Aij|.
Hnx2=H0x2+b Randx2,

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