Abstract

Access to the spatial derivatives of an optical wave field can be used to enhance edge detection, focusing, and holographic imaging. It was recently shown that, by using digital holographic techniques, the Laplacian of an object field can be extracted. Here it is demonstrated that equivalent results can be found using two holograms captured at either two distances or with two appropriately related wavelengths. Experimental and numerical results confirming the theoretical analyses are presented. The proposed two-wavelength-based system requires no mechanical repositioning of the object and is shown to provide superior performance.

© 2010 Optical Society of America

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References

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2010

Y. Han and Q. Yue, Opt. Commun. 283, 929 (2010).
[CrossRef]

2008

2004

Y. Zhang, G. Pedrini, W. Osten, and H. J. Tiziani, Opt. Lett. 29, 1787 (2004).
[CrossRef] [PubMed]

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, Ultramicroscopy 102, 37 (2004).
[CrossRef] [PubMed]

2000

J. B. Tiller, A. Barty, D. Paganin, and K. A. Nugent, Opt. Commun. 183, 7 (2000).
[CrossRef]

1997

1994

1991

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

Barty, A.

J. B. Tiller, A. Barty, D. Paganin, and K. A. Nugent, Opt. Commun. 183, 7 (2000).
[CrossRef]

Beleggia, M.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, Ultramicroscopy 102, 37 (2004).
[CrossRef] [PubMed]

Gopinathan, U.

Guo, C.-S.

Han, Y.

Y. Han and Q. Yue, Opt. Commun. 283, 929 (2010).
[CrossRef]

Jüptner, W.

Lu, L.-L.

Nugent, K. A.

J. B. Tiller, A. Barty, D. Paganin, and K. A. Nugent, Opt. Commun. 183, 7 (2000).
[CrossRef]

Osten, W.

Paganin, D.

J. B. Tiller, A. Barty, D. Paganin, and K. A. Nugent, Opt. Commun. 183, 7 (2000).
[CrossRef]

Pedrini, G.

Ryle, J. P.

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

Schnars, U.

Schofield, M. A.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, Ultramicroscopy 102, 37 (2004).
[CrossRef] [PubMed]

Sheridan, J. T.

Situ, G.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

Tiller, J. B.

J. B. Tiller, A. Barty, D. Paganin, and K. A. Nugent, Opt. Commun. 183, 7 (2000).
[CrossRef]

Tiziani, H. J.

Volkov, V. V.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, Ultramicroscopy 102, 37 (2004).
[CrossRef] [PubMed]

Wei, G.-X.

Yamaguchi, I.

Yue, Q.

Y. Han and Q. Yue, Opt. Commun. 283, 929 (2010).
[CrossRef]

Yue, Q.-Y.

Yue, S.-J.

Zhang, T.

Zhang, Y.

Zhu, Y.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, Ultramicroscopy 102, 37 (2004).
[CrossRef] [PubMed]

Appl. Opt.

Opt. Commun.

J. B. Tiller, A. Barty, D. Paganin, and K. A. Nugent, Opt. Commun. 183, 7 (2000).
[CrossRef]

Y. Han and Q. Yue, Opt. Commun. 283, 929 (2010).
[CrossRef]

Opt. Lett.

Ultramicroscopy

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, Ultramicroscopy 102, 37 (2004).
[CrossRef] [PubMed]

Other

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Schematic of the experimental setup.

Fig. 2
Fig. 2

In (a), a horizontal 1 mm bar is present in the reconstructed image | o ( x , y , z = 0 , λ ) | and, in (b), | x y 2 o ( x , y , z = 0 , λ ) | , calculated directly from a single hologram using MATLAB. (c) Δ H z and (d) the corresponding magnitude of the resulting Laplacian differential reconstruction at z = 160 mm [see Eq. (6)] with z + Δ z = 165 mm and λ = 632.8 nm . (e) Δ H λ with z = 160 nm and λ = 632.8 nm and λ + Δ λ = 652.5 nm . (f) Magnitudes of Laplacian differential reconstructions: (d) z = 160 ; see Eq. (10). (g) Magnitude and (h) phase, calculated from a difference hologram Δ H z λ by combining both methods.

Fig. 3
Fig. 3

In (a), | o ( x , y , z = 0 , λ ) | and, in (b), | x y 2 o ( x , y , z = 0 , λ ) | , calculated directly from a single hologram using MATLAB. (c) Two-wavelength difference hologram, Δ H λ , with z = 160 mm , and λ = 632.8 nm and λ + Δ λ = 652.5 nm , while (d) shows the resulting reconstructed Laplacian magnitude from (c). (e) Magnitude of the two-plane Laplacian differential reconstruction at z = 160 mm [see Eq. (6)], with z + Δ z = 165 mm and λ = 632.8 nm . Mean normalized pixel values from Figs. 2c, 2d, 2e are replotted and shown in (f). Note the symmetry and narrow peaks indicating the edges, as well as the low values in the center, in (d).

Equations (10)

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H ( x , y , z , λ ) DC = o ( x , y , z , λ ) r * ( x , y , z , λ ) object + o * ( x , y , z , λ ) r ( x , y , z , λ ) twin .
o ( x , y , z , λ ) = exp ( + i 2 π λ z ) I 1 { O ( ξ , η , 0 , λ ) exp [ i π z λ ( ξ 2 + η 2 ) ] } .
Δ H z = r ( x , y , z , λ ) [ o * ( x , y , z + Δ z , λ ) o * ( x , y , z , λ ) ] + r * ( x , y , z , λ ) [ o ( x , y , z + Δ z , λ ) o ( x , y , z , λ ) ] .
Δ H z = Δ z r ( x , y , z , λ ) I 1 { + i π λ ( ξ 2 + η 2 ) O * ( ξ , η , 0 , λ ) exp [ + i π λ z ( ξ 2 + η 2 ) ] } + Δ z r * ( x , y , z , λ ) I 1 { i π λ ( ξ 2 + η 2 ) O ( ξ , η , 0 , λ ) exp [ i π λ z ( ξ 2 + η 2 ) ] } .
P z ( Δ H z ) = Δ z r ( x , y , z , λ ) I 1 { + i π λ ( ξ 2 + η 2 ) O * ( ξ , η , 0 , λ ) exp [ + i 2 π λ z ( ξ 2 + η 2 ) ] } + Δ z r * ( x , y , z , λ ) I 1 { i π λ ( ξ 2 + η 2 ) O ( ξ , η , 0 , λ ) } .
P z ( Δ H z ) = i λ Δ z 4 π r * ( x , y , z , λ ) x y 2 o ( x , y , 0 , λ ) twin .
exp ( + i 2 π λ + Δ λ z ) I 1 { O ( ξ , η , 0 , λ + Δ λ ) exp [ i π z ( λ + Δ λ ) ( ξ 2 + η 2 ) ] } ,
H ( x , y , z , λ ) D C = r * ( x , y , z , λ ) I 1 { O ( ξ , η , 0 , λ ) exp [ i π z λ ( ξ 2 + η 2 ) ] } + r ( x , y , z , λ ) I 1 { O * ( ξ , η , 0 , λ ) exp [ + i π z λ ( ξ 2 + η 2 ) ] } .
Δ H λ = r * ( x , y , z , λ ) Δ λ I 1 { i π z ( ξ 2 + η 2 ) O ( ξ , η , 0 , λ ) exp [ i π z λ ( ξ 2 + η 2 ) ] } + r ( x , y , z , λ ) Δ λ I 1 { + i π z ( ξ 2 + η 2 ) O * ( ξ , η , 0 , λ ) exp [ + i π z λ ( ξ 2 + η 2 ) ] } .
P z ( Δ H λ ) = i z Δ λ 4 π r * ( x , y , z , λ ) x y 2 o ( x , y , 0 , λ ) twin .

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