Abstract

We present an effective method for the pixel-size-maintained reconstruction of images on arbitrarily tilted planes in digital holography. The method is based on the plane wave expansion of the diffraction wave fields and the three-axis rotation of the wave vectors. The images on the tilted planes are reconstructed without loss of the frequency contents of the hologram and have the same pixel sizes. Our method shows good results in the extreme cases of large tilting angles and in the region closer than the paraxial case. The effectiveness of the method is demonstrated by both simulation and experiment.

© 2008 Optical Society of America

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2007 (1)

J. W. Kang and C. K. Hong, “Phase-contrast microscopy by in-line phase-shifting digital holography: shape measurement of a titanium pattern with nanometer axial resolution,” Opt. Eng. 46, 040506 (2007).
[CrossRef]

2006 (5)

2005 (4)

2004 (4)

P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, “Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms,” Opt. Lett. 29, 854-856 (2004).
[CrossRef] [PubMed]

F. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett. 29, 1668-1670 (2004).
[CrossRef] [PubMed]

H. Y. Yun, S. J. Jeong, J. W. Kang, and C. K. Hong, “3-dimensional micro-structure inspection by phase-shifting digital holography,” Key Eng. Mater. 270-273, 756-761 (2004).
[CrossRef]

P. Ferraro, S. De Nicola, A. Finizio, G. Pierattini, and G. Coppola, “Recovering image resolution in reconstructing digital off-axis holograms by Fresnel-transform method,” Appl. Phy. Lett. 85, 2709-2711 (2004).
[CrossRef]

2003 (2)

2002 (1)

2001 (2)

2000 (2)

1999 (1)

1998 (1)

A. F. Ware, “Fast approximate Fourier transforms for irregularly spaced data,” SIAM Rev. 40, 838-856 (1998).
[CrossRef]

1997 (1)

1995 (1)

A. Dutt and V. Rokhlin, “Fast Fourier transforms for nonequispaced data, II,” Appl. Comput. Harmon. Anal. 2, 85-100(1995).
[CrossRef]

1994 (1)

Alfieri, D.

An, Y.

Aspert, N.

Asundi, A. K.

Benkouider, A.

Bevilacqua, F.

Bourquin, S.

Cai, L.

Charrière, F.

Coëtmellec, S.

Colomb, T.

Coppola, G.

P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, “Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms,” Opt. Lett. 29, 854-856 (2004).
[CrossRef] [PubMed]

P. Ferraro, S. De Nicola, A. Finizio, G. Pierattini, and G. Coppola, “Recovering image resolution in reconstructing digital off-axis holograms by Fresnel-transform method,” Appl. Phy. Lett. 85, 2709-2711 (2004).
[CrossRef]

Cuche, E.

De Nicola, S.

Depeursinge, C.

Dutt, A.

A. Dutt and V. Rokhlin, “Fast Fourier transforms for nonequispaced data, II,” Appl. Comput. Harmon. Anal. 2, 85-100(1995).
[CrossRef]

Emery, Y.

Ferraro, P.

Finizio, A.

Hong, C. K.

J. W. Kang and C. K. Hong, “Phase-contrast microscopy by in-line phase-shifting digital holography: shape measurement of a titanium pattern with nanometer axial resolution,” Opt. Eng. 46, 040506 (2007).
[CrossRef]

H. Y. Yun, S. J. Jeong, J. W. Kang, and C. K. Hong, “3-dimensional micro-structure inspection by phase-shifting digital holography,” Key Eng. Mater. 270-273, 756-761 (2004).
[CrossRef]

Javidi, B.

Jeong, S. J.

H. Y. Yun, S. J. Jeong, J. W. Kang, and C. K. Hong, “3-dimensional micro-structure inspection by phase-shifting digital holography,” Key Eng. Mater. 270-273, 756-761 (2004).
[CrossRef]

Jüptner, W.

Kang, J. W.

J. W. Kang and C. K. Hong, “Phase-contrast microscopy by in-line phase-shifting digital holography: shape measurement of a titanium pattern with nanometer axial resolution,” Opt. Eng. 46, 040506 (2007).
[CrossRef]

H. Y. Yun, S. J. Jeong, J. W. Kang, and C. K. Hong, “3-dimensional micro-structure inspection by phase-shifting digital holography,” Key Eng. Mater. 270-273, 756-761 (2004).
[CrossRef]

Kato, J.-i.

Kim, M. K.

Kühn, J.

Lebrun, D.

Magistretti, P. J.

Malek, M.

Marian, A.

Marquet, P.

Massig, J. H.

Matsushima, K.

Miao, J.

Mizuno, J.

Montfort, F.

Ohta, S.

Ohzu, H.

Peng, X.

Pierattini, G.

Rappaz, B.

Rokhlin, V.

A. Dutt and V. Rokhlin, “Fast Fourier transforms for nonequispaced data, II,” Appl. Comput. Harmon. Anal. 2, 85-100(1995).
[CrossRef]

Schimmel, H.

Schnars, U.

Stadelmaier, A.

Stern, A.

Takaki, Y.

Ware, A. F.

A. F. Ware, “Fast approximate Fourier transforms for irregularly spaced data,” SIAM Rev. 40, 838-856 (1998).
[CrossRef]

Wyrowski, F.

Xu, L.

Yamaguchi, I.

Yaroslavsky, L. P.

Yu, L.

Yun, H. Y.

H. Y. Yun, S. J. Jeong, J. W. Kang, and C. K. Hong, “3-dimensional micro-structure inspection by phase-shifting digital holography,” Key Eng. Mater. 270-273, 756-761 (2004).
[CrossRef]

Zhang, F.

Zhang, T.

Appl. Comput. Harmon. Anal. (1)

A. Dutt and V. Rokhlin, “Fast Fourier transforms for nonequispaced data, II,” Appl. Comput. Harmon. Anal. 2, 85-100(1995).
[CrossRef]

Appl. Opt. (4)

Appl. Phy. Lett. (1)

P. Ferraro, S. De Nicola, A. Finizio, G. Pierattini, and G. Coppola, “Recovering image resolution in reconstructing digital off-axis holograms by Fresnel-transform method,” Appl. Phy. Lett. 85, 2709-2711 (2004).
[CrossRef]

J. Opt. Soc. Am. A (3)

Key Eng. Mater. (1)

H. Y. Yun, S. J. Jeong, J. W. Kang, and C. K. Hong, “3-dimensional micro-structure inspection by phase-shifting digital holography,” Key Eng. Mater. 270-273, 756-761 (2004).
[CrossRef]

Opt. Commun. (1)

L. Yu and M. K. Kim, “Variable tomographic scanning with wavelength scanning digital interference holography.” Opt. Commun. 260, 462-468 (2006).
[CrossRef]

Opt. Eng. (1)

J. W. Kang and C. K. Hong, “Phase-contrast microscopy by in-line phase-shifting digital holography: shape measurement of a titanium pattern with nanometer axial resolution,” Opt. Eng. 46, 040506 (2007).
[CrossRef]

Opt. Express (6)

Opt. Lett. (7)

SIAM Rev. (1)

A. F. Ware, “Fast approximate Fourier transforms for irregularly spaced data,” SIAM Rev. 40, 838-856 (1998).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Geometrical configuration of the image and the hologram planes. The tilted image plane is defined by the angles ϕ and θ.

Fig. 2
Fig. 2

Geometrical configuration of the transformed wave-vector space: the dashed-square mark is the original wave-vector space shifted by k sin θ in the k x direction, and the diamond shapes marked by dotted–dashed and solid lines are the wave-vector spaces transformed by two- and three-axis rotations, respectively.

Fig. 3
Fig. 3

Geometry of two point objects and the hologram and the image planes.

Fig. 4
Fig. 4

(a) Geometry of image reconstruction: the second row corresponds to the normalized amplitudes reconstructed (b) with the full inclination factor and (c) with the Fresnel-approximated one at θ = 90 ° , and (d) their difference. (e) and (f) are the phase maps corresponding to (b) and (c), respectively. (g) is the difference between (e) and (f) in radians.

Fig. 5
Fig. 5

Difference between the images of the point object reconstructed by using the inclination factors with and without the Fresnel approximation for various tilting angles.

Fig. 6
Fig. 6

(a) In-line digital holography setup. BS, beam splitter; M1, mirror with a piezo-electric transducer; M2, mirror; SF, spatial filter; L, collimating lens. (b) Diffraction pattern of the tilted 1951 USAF resolution target on the CCD plane.

Fig. 7
Fig. 7

Transformed wave vectors (log scale) with (a)  ϕ = 0 ° , θ = 0 ° , and ϕ = 0 ° , (b)  ϕ = 45 ° , θ = 40 ° , and ϕ = 0 ° , and (c)  ϕ = 45 ° , θ = 40 ° , and ϕ = 45 ° .

Fig. 8
Fig. 8

Reconstructed images of the tilted resolution target ( 640 × 480 pixels). (a) Image on the plane parallel to the CCD at z = 2.20 cm reconstructed with the ASM. Images on the tilted plane at z = 2.42 cm reconstructed from (b) the whole area and (c) only the dashed-square area of Fig. 5(b). (d) Image reconstructed by our method.

Equations (17)

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U z ( k x , k y ) = U 0 ( k x , k y ) · P z ( k z ) ,
P z ( k z ) = exp ( i k z z )
k z = ( k 2 k x 2 k y 2 ) 1 / 2 .
u z ( x , y ) = F 1 [ U 0 ( k x , k y ) · P z ( k z ) ] .
k X = k x cos θ cos ϕ + k y cos θ sin ϕ + k z sin θ , k Y = k x sin ϕ + k y cos ϕ , k Z = k x sin θ cos ϕ k y sin θ sin ϕ + k z cos θ .
U z , tilted ( k X , k Y ) = U z ( k x cos θ cos ϕ + k y cos θ sin ϕ + k z sin θ , k x sin ϕ + k y cos ϕ ) .
i ( z ˆ , Z ˆ , k ) = | Z ˆ · k z ˆ · k | 1 / 2 = | k x sin θ cos ϕ k y sin θ sin ϕ + k z cos θ k z | 1 / 2
i ( z ˆ , Z ˆ , k ) cos θ k x sin θ cos ϕ + k y sin θ sin ϕ 2 k z cos θ
U z , tilted ( k X , k Y ) = i ( z ˆ , Z ˆ , k ) · U z ( k x cos θ cos ϕ + k y cos θ sin ϕ + k z sin θ , k x sin ϕ + k y cos ϕ ) .
k X k X k sin θ = k x cos θ cos ϕ + k y cos θ sin ϕ + k z sin θ k sin θ ,
Δ v X Δ v x | cos θ cos ϕ | + Δ v y | cos θ sin ϕ | , Δ v Y = Δ v x | sin ϕ | + Δ v y | cos ϕ |
Δ v X Δ v z sin θ , Δ v Y = Δ ν x | sin ϕ | + Δ v y | cos ϕ |
k X = k X cos ϕ + k Y sin ϕ , k Y = k X sin ϕ + k Y cos ϕ , k Z = k Z
ϕ = [ tan 1 ( k Y , max k X , corr ) + π 4 ]
ϕ = [ tan 1 ( k Y , max k X , corr ) π 4 ]
u z , tilted ( X , Y ) = F 1 [ T 2 [ T 1 [ F [ u 0 ( x , y ) ] · P z ( k z ) ] ] ] = F 1 [ T 2 [ T 1 [ U 0 ( k x , k y ) · P z ( k z ) ] ] ] = F 1 [ T 2 [ T 1 [ U z ( k x , k y ) ] ] ] = F 1 [ T 2 [ U z , tilted ( k X + k sin θ , k Y ) ] ] = exp [ i k sin θ ( X cos ϕ Y sin ϕ ) ] · F 1 [ U z , tilted ( k X , k Y ) ] .
u z , tilted ( m δ X , n δ Y ) = exp [ i k sin θ ( m δ X cos ϕ n δ Y sin ϕ ) ] p , q U z , tilted ( k X ( p , q ) , k Y ( p , q ) ) exp [ i ( k X ( p , q ) m δ X + k Y ( p , q ) n δ Y ) ] ,

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