Abstract

We describe the recording conditions that, together with the appropriate numerical reconstruction process, permit high-lateral-resolution reconstruction of in-line digital holograms. By high resolution, we mean a resolution that is beyond the Nyquist frequency, which is achieved by common methods. The proposed method is based on a previously reported generalized sampling theory that presents the conditions to precisely reconstruct fields that in certain cases may be sampled with a sampling rate lower than the Nyquist rate. We examine the hologram-recording process in the Wigner space. On the basis of this analysis, we demonstrate a simple high-resolution numerical reconstruction method.

© 2006 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  12. M. Sypek, "Light propagation in the Fresnel region. New numerical approach," Opt. Commun. 116, 43-48 (1995).
    [CrossRef]
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    [CrossRef]
  14. A. Stern and B. Javidi, "Analysis of practical sampling and reconstruction from Fresnel fields," Opt. Eng. 43, 239-250 (2004).
    [CrossRef]
  15. A. Stern and B. Javidi, "Sampling in the light of Wigner distribution," J. Opt. Soc. Am. A 21, 360-366 (2004); A. Stern and B. Javidi, "Sampling in the light of Wigner distribution," J. Opt. Soc. Am. A errata 21, 2038 (2004).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  25. A. Gori, "Fresnel transform and sampling theorem," Opt. Commun. 39, 293-297 (1981).
    [CrossRef]
  26. L. Onural, "Sampling of the diffraction field," Appl. Opt. 39, 5929-5935 (2000).
    [CrossRef]
  27. J. M. Whittaker, "The Fourier theory of cardinal functions," Proc. Edinb. Math. Soc. 1, 169-176 (1929).
    [CrossRef]
  28. M. Unser, "Sampling—50 years after Shannon," Proc. IEEE 88, 569-587 (2000).
    [CrossRef]
  29. D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty, I," Bell Syst. Tech. J. 40, 43-64 (1961).
  30. S. Mann and S. Haykin, "The chirplet transform: physical considerations," IEEE Trans. Signal Process. 43, 2745-2761 (1995).
    [CrossRef]
  31. K. Grochenig, Foundation of Time-frequency Analysis (Birkhäuser, 2001).
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2004 (3)

A. Stern and B. Javidi, "Analysis of practical sampling and reconstruction from Fresnel fields," Opt. Eng. 43, 239-250 (2004).
[CrossRef]

A. Stern and B. Javidi, "Sampling in the light of Wigner distribution," J. Opt. Soc. Am. A 21, 360-366 (2004); A. Stern and B. Javidi, "Sampling in the light of Wigner distribution," J. Opt. Soc. Am. A errata 21, 2038 (2004).
[CrossRef]

A. Stern and B. Javidi, "General sampling theorem and application in digital holography," in Proc. SPIE 5557, 110-123 (2004).
[CrossRef]

2003 (1)

D. Mas, J. Perez, C. Hernandez, C. Vazquez, J. J. Miret, and C. Illueca, "Fast numerical calculation of Fresnel patterns in convergent systems," Opt. Commun. 227, 245-258 (2003).
[CrossRef]

2002 (1)

2001 (3)

2000 (5)

1998 (2)

1997 (2)

1996 (2)

1995 (3)

W. W. Smith and J. M. Smith, Handbook of Real-Time Fast Fourier Transforms (IEEE, 1995).
[CrossRef]

M. Sypek, "Light propagation in the Fresnel region. New numerical approach," Opt. Commun. 116, 43-48 (1995).
[CrossRef]

S. Mann and S. Haykin, "The chirplet transform: physical considerations," IEEE Trans. Signal Process. 43, 2745-2761 (1995).
[CrossRef]

1994 (1)

1993 (1)

1992 (1)

1981 (1)

A. Gori, "Fresnel transform and sampling theorem," Opt. Commun. 39, 293-297 (1981).
[CrossRef]

1980 (2)

M. J. Bastiaans, "Wigner distribution function and its application to first-order optics," J. Opt. Soc. Am. 69, 1710-1716 (1980).
[CrossRef]

A. Riazi, O. P. Gandhi, and D. A. Christensen, "Efficient FFT computation of plano-plano interferometer modes for a wide range of Fresnel numbers," Opt. Acta 27, 529-535 (1980).
[CrossRef]

1975 (1)

1966 (1)

1961 (1)

D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty, I," Bell Syst. Tech. J. 40, 43-64 (1961).

1946 (1)

D. Gabor, Theory of communication, J. Inst. Electr. Eng., Part 3 93, 429-457 (1946).

1932 (1)

E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
[CrossRef]

1929 (1)

J. M. Whittaker, "The Fourier theory of cardinal functions," Proc. Edinb. Math. Soc. 1, 169-176 (1929).
[CrossRef]

Arrizon, V.

Asundi, A.

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

Bastiaans, M. J.

Castro, A.

Castro, M. A.

Catrysse, P. B.

Christensen, D. A.

A. Riazi, O. P. Gandhi, and D. A. Christensen, "Efficient FFT computation of plano-plano interferometer modes for a wide range of Fresnel numbers," Opt. Acta 27, 529-535 (1980).
[CrossRef]

Dorsch, R. G.

Ferreira, C.

Frauel, Y.

Gabor, D.

D. Gabor, Theory of communication, J. Inst. Electr. Eng., Part 3 93, 429-457 (1946).

Gandhi, O. P.

A. Riazi, O. P. Gandhi, and D. A. Christensen, "Efficient FFT computation of plano-plano interferometer modes for a wide range of Fresnel numbers," Opt. Acta 27, 529-535 (1980).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Gori, A.

A. Gori, "Fresnel transform and sampling theorem," Opt. Commun. 39, 293-297 (1981).
[CrossRef]

Grochenig, K.

K. Grochenig, Foundation of Time-frequency Analysis (Birkhäuser, 2001).

Haykin, S.

S. Mann and S. Haykin, "The chirplet transform: physical considerations," IEEE Trans. Signal Process. 43, 2745-2761 (1995).
[CrossRef]

Hernandez, C.

D. Mas, J. Perez, C. Hernandez, C. Vazquez, J. J. Miret, and C. Illueca, "Fast numerical calculation of Fresnel patterns in convergent systems," Opt. Commun. 227, 245-258 (2003).
[CrossRef]

Illueca, C.

D. Mas, J. Perez, C. Hernandez, C. Vazquez, J. J. Miret, and C. Illueca, "Fast numerical calculation of Fresnel patterns in convergent systems," Opt. Commun. 227, 245-258 (2003).
[CrossRef]

Javidi, B.

Juptner, W.

Kopeika, S.

S. Kopeika, A System Engineering Approach to Imaging (SPIE, 1998).

Lohmann, A. W.

Lukosz, W.

Mann, S.

S. Mann and S. Haykin, "The chirplet transform: physical considerations," IEEE Trans. Signal Process. 43, 2745-2761 (1995).
[CrossRef]

Mas, D.

D. Mas, J. Perez, C. Hernandez, C. Vazquez, J. J. Miret, and C. Illueca, "Fast numerical calculation of Fresnel patterns in convergent systems," Opt. Commun. 227, 245-258 (2003).
[CrossRef]

Matoba, O.

Mendlovic, D.

Miao, K.

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

Miret, J. J.

D. Mas, J. Perez, C. Hernandez, C. Vazquez, J. J. Miret, and C. Illueca, "Fast numerical calculation of Fresnel patterns in convergent systems," Opt. Commun. 227, 245-258 (2003).
[CrossRef]

Ojeda-Castaneda, J.

Ojeda-Castañeda, J.

Onural, L.

Perez, J.

D. Mas, J. Perez, C. Hernandez, C. Vazquez, J. J. Miret, and C. Illueca, "Fast numerical calculation of Fresnel patterns in convergent systems," Opt. Commun. 227, 245-258 (2003).
[CrossRef]

Pollak, H. O.

D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty, I," Bell Syst. Tech. J. 40, 43-64 (1961).

Riazi, A.

A. Riazi, O. P. Gandhi, and D. A. Christensen, "Efficient FFT computation of plano-plano interferometer modes for a wide range of Fresnel numbers," Opt. Acta 27, 529-535 (1980).
[CrossRef]

Schnars, U.

Siegman, A. E.

Slepian, D.

D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty, I," Bell Syst. Tech. J. 40, 43-64 (1961).

Smith, J. M.

W. W. Smith and J. M. Smith, Handbook of Real-Time Fast Fourier Transforms (IEEE, 1995).
[CrossRef]

Smith, W. W.

W. W. Smith and J. M. Smith, Handbook of Real-Time Fast Fourier Transforms (IEEE, 1995).
[CrossRef]

Stern, A.

A. Stern and B. Javidi, "Sampling in the light of Wigner distribution," J. Opt. Soc. Am. A 21, 360-366 (2004); A. Stern and B. Javidi, "Sampling in the light of Wigner distribution," J. Opt. Soc. Am. A errata 21, 2038 (2004).
[CrossRef]

A. Stern and B. Javidi, "Analysis of practical sampling and reconstruction from Fresnel fields," Opt. Eng. 43, 239-250 (2004).
[CrossRef]

A. Stern and B. Javidi, "General sampling theorem and application in digital holography," in Proc. SPIE 5557, 110-123 (2004).
[CrossRef]

Sypek, M.

M. Sypek, "Light propagation in the Fresnel region. New numerical approach," Opt. Commun. 116, 43-48 (1995).
[CrossRef]

Szilkas, E. A.

Tajahuerce, E.

Unser, M.

M. Unser, "Sampling—50 years after Shannon," Proc. IEEE 88, 569-587 (2000).
[CrossRef]

Vazquez, C.

D. Mas, J. Perez, C. Hernandez, C. Vazquez, J. J. Miret, and C. Illueca, "Fast numerical calculation of Fresnel patterns in convergent systems," Opt. Commun. 227, 245-258 (2003).
[CrossRef]

Wandell, B. A.

Whittaker, J. M.

J. M. Whittaker, "The Fourier theory of cardinal functions," Proc. Edinb. Math. Soc. 1, 169-176 (1929).
[CrossRef]

Wigner, E.

E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Xu, L.

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

Yamaguchi, I.

Zalevsky, Z.

Zhang, T.

Appl. Opt. (5)

Bell Syst. Tech. J. (1)

D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty, I," Bell Syst. Tech. J. 40, 43-64 (1961).

IEEE Trans. Signal Process. (1)

S. Mann and S. Haykin, "The chirplet transform: physical considerations," IEEE Trans. Signal Process. 43, 2745-2761 (1995).
[CrossRef]

J. Inst. Electr. Eng., Part 3 (1)

D. Gabor, Theory of communication, J. Inst. Electr. Eng., Part 3 93, 429-457 (1946).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (7)

Opt. Acta (1)

A. Riazi, O. P. Gandhi, and D. A. Christensen, "Efficient FFT computation of plano-plano interferometer modes for a wide range of Fresnel numbers," Opt. Acta 27, 529-535 (1980).
[CrossRef]

Opt. Commun. (3)

M. Sypek, "Light propagation in the Fresnel region. New numerical approach," Opt. Commun. 116, 43-48 (1995).
[CrossRef]

D. Mas, J. Perez, C. Hernandez, C. Vazquez, J. J. Miret, and C. Illueca, "Fast numerical calculation of Fresnel patterns in convergent systems," Opt. Commun. 227, 245-258 (2003).
[CrossRef]

A. Gori, "Fresnel transform and sampling theorem," Opt. Commun. 39, 293-297 (1981).
[CrossRef]

Opt. Eng. (2)

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

A. Stern and B. Javidi, "Analysis of practical sampling and reconstruction from Fresnel fields," Opt. Eng. 43, 239-250 (2004).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. (1)

E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Proc. Edinb. Math. Soc. (1)

J. M. Whittaker, "The Fourier theory of cardinal functions," Proc. Edinb. Math. Soc. 1, 169-176 (1929).
[CrossRef]

Proc. IEEE (1)

M. Unser, "Sampling—50 years after Shannon," Proc. IEEE 88, 569-587 (2000).
[CrossRef]

Proc. SPIE (1)

A. Stern and B. Javidi, "General sampling theorem and application in digital holography," in Proc. SPIE 5557, 110-123 (2004).
[CrossRef]

Other (4)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

S. Kopeika, A System Engineering Approach to Imaging (SPIE, 1998).

W. W. Smith and J. M. Smith, Handbook of Real-Time Fast Fourier Transforms (IEEE, 1995).
[CrossRef]

K. Grochenig, Foundation of Time-frequency Analysis (Birkhäuser, 2001).

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Figures (8)

Fig. 1
Fig. 1

General schematic description of digital holography. FRT, Fresnel transform.

Fig. 2
Fig. 2

Outline of an in-line phase-shifting digital holography system.

Fig. 3
Fig. 3

Digital hologram formation in Wigner space. (a) The space–bandwidth product [ SW 0 ( x , ν ) ] of the object u 0 ( x ) . (b) The space–bandwidth product SW z ( x , ν ) of the object CFA u z ( x ) at the hologram plane. SW is sheared in the x direction; therefore the hologram sensor size needs to be larger than B x + λ z B ν in order to capture the entire information. (c) The effect of sampling of u z ( x ) in the Wigner domain. MLB, maximum local bandwidth.

Fig. 4
Fig. 4

Direct digital reconstruction process. (a) The acceptance space–bandwidth product [ SW D Y ( x , ν ) ] of the digital reconstruction system. (b) The part of the WD of the sampled signal [Fig. 3c] processed numerically. (c) The space–bandwidth product of the reconstructed signal SW u ̂ .

Fig. 5
Fig. 5

Reconstruction by directly inverse Fresnel transforming a signal that does not fulfill the Nyquist criterion [ B ν > 1 Δ ] . (a) The space–bandwidth product SW z of the sampled signal. (b) The space–bandwidth product SW z D processed digitally. (c) The space–bandwidth product of the reconstructed signal. It can be seen that the SW of the original signal is not recovered.

Fig. 6
Fig. 6

High-resolution reconstruction. (a) Space–bandwidth product SW z of the sampled signal. The sampling rate is smaller than the total bandwidth ( 1 Δ < B ν ) but is larger than the MLB; therefore the generalized sampling condition [expression (12)] is fulfilled. (b) The space–bandwidth product processed digitally after upsampling with a ratio r = 2 . The SW Y D is twice larger than in Fig. 5; therefore the high-frequency components are preserved in the reconstruction. (c) After the IFRT, a masking in the range x B x 2 is performed to reject first-order artifacts.

Fig. 7
Fig. 7

Digital holography simulation results. (a) Chessboard target. The spatial frequency in the lower half of the target is too high to fulfill the Nyquist criterion but does fulfill the condition in Eq. (13). (b) The absolute value of the object CFA at the sensor plane. (c) Reconstruction by performing a direct IFRT. The lower half of the target is totally distorted by aliasing. (d) High-resolution algorithm reconstruction with the proposed algorithm.

Fig. 8
Fig. 8

Shaded area represents the SW ( x , ν ) of a continuous signal f ( x ) . The length of the signal is B x , the bandwidth is B ν , and MLB denotes the maximum local bandwidth. The SW ( x , ν ) is enclosed by a parallelogram x B x 2 , ν ¯ ( x ) ( MLB 2 ) ν ν ¯ ( x ) ( MLB 2 ) . f ( x ) can be represented by N B x MLB coefficients.

Tables (1)

Tables Icon

Table 1 Comparison between the SNRs Obtained Using the Direct Reconstruction Method and the Proposed Method

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

u z ( x ) = 1 j λ z u ( ξ ) exp [ j π λ z ( x ξ ) 2 ] d ξ ,
u ̂ = F 1 { F { u z } Q z ( ν ) } ,
Q z ( ν ) = exp ( j π λ z ν 2 ) ,
q ( x ) = 1 j λ z exp [ j π λ ( z ) x 2 ] .
W u 0 ( x , ν ) = u 0 ( x + x 2 ) u 0 * ( x x 2 ) exp ( j 2 π x ν ) d x .
W u z ( x , ν ) = W u 0 ( x λ z ν , ν ) .
L H = B x + λ z B ν ,
W u x ( x , ν ) = 1 2 Δ n k W u z ( x n Δ , ν k 1 2 Δ ) δ ( x n Δ ) + 1 2 Δ n k ( 1 ) k W u z ( x n Δ Δ 2 , ν k 1 2 Δ ) δ ( x ( n + 1 2 ) Δ ) ,
u z [ n ] = u z ( n Δ ) , n = 2 , 1 , 0 , 1 , 2 , , L 2 n Δ < L 2 ,
1 2 Δ < ν < 1 2 Δ .
SW z D = SW Y D SW z ,
SW Y D ( x , ν ) = { 1 x N Δ 2 , ν 1 ( 2 Δ ) 0 otherwise } .
1 Δ MLB .
1 Δ B x λ z .
r max ( B ν 1 Δ , 1 ) ;
u z [ m ] { u z ( n Δ ) m r = n , m = 1 , 01 , 2 , , r N m r N 0 otherwise } .
r max ( B ν 1 Δ , 1 ) .
λ Δ z B x ( z ) .
z o Δ λ B x _ max ,
SNR = 20 log 10 u 0 u 0 u ̂ z ,
f ( x ) = k Z c ( k ) ϕ k ( x ) ,
f ( x ) = k = 1 N c ( k ) ϕ k ( x ) ,
N B x MLB .
N B x MLB .
u z ( x ) = n u z [ n ] δ ( x n Δ ) ,
W u z ( x , ν ) = u z ( x + x 2 ) u z * ( x x 2 ) exp ( j 2 π x ν ) d x ,
W u z ( x , ν ) = [ n u z [ n ] δ ( x + x 2 n Δ ) ] [ n u z * [ n ] δ ( x x 2 n Δ ) ] exp ( j 2 π x ν ) d x = n , n u z [ n ] u z * [ n ] exp ( 2 j π ν ( n n ) Δ ) δ ( x ( n + n ) Δ 2 ) ,
W u z ( x , ν ) = n , n u z [ n ] ( u z [ n ] ) * exp ( 2 j π ν ( n n ) Δ r ) δ ( x ( n + n ) Δ 2 r ) .
W u z ( x , ν ) = n , n u z [ n ] ( u z [ n ] ) * exp ( 2 j π ν ( n n ) Δ ) δ ( x ( n + n ) Δ 2 ) ,

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