Abstract

Acceptable signal recovery of the band-pass signals typically used in the off-axis digital holography systems is possible in the undersampling conditions. A typical system is considered in which the angle between two beams represents a variable parameter. For the given signal bandwidth and experimental conditions the hologram reconstruction is constrained by the sampling frequency of the array photo-detector. Reconstructions from the undersampled digital holograms are analyzed both theoretically and experimentally. It is shown how increasing the angle values beyond the Nyquist limits leads to repeatedly folding and inverting the reconstructed object image until the fading of the image. The phase point at the image fading and the non-overlapping intervals for correctly preserving the useful information are defined and evaluated. Amplitude distributions are analyzed on the example of the time-averaged holograms acquired for an oscillating membrane. Based on removing the zeroth-order reconstruction term, significant extensions of these intervals are also demonstrated.

© 2009 Optical Society of America

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    [CrossRef] [PubMed]

2008

2007

2006

2005

2004

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

N. Demoli and D. Vukicevic, "Detection of hidden stationary deformations of vibrating surfaces by use of time-averaged digital holographic interferometry," Opt. Lett. 29, 1423-2425 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=ol-29-20-2423.
[CrossRef]

N. Demoli, D. Vukicevic, and M. Torzynski, "Time-averaged holographic interferometry using subtraction digital holography," Proc. SPIE 5457, 643-650 (2004).
[CrossRef]

P. Perez and A. Santos, "Undersampling to acquire nuclear magnetic resonance images," Med. Eng. Phys. 26, 523-529 (2004).
[CrossRef] [PubMed]

2003

2002

T. Mishina, M. Okui, and F. Okano, "Viewing-zone enlargement method for sampled hologram that uses high-order diffraction," Appl. Opt. 41, 1489-1499 (2002).
[CrossRef] [PubMed]

T. Kreis, "Frequency analysis of digital holography," Opt. Eng. 41, 771-778 (2002).
[CrossRef]

J. H. Massig, "Digital off-axis holography with a synthetic aperture," Opt. Lett. 27, 2179-2181 (2002).
[CrossRef]

C. Liu, Z. Liu, F. Bo, Y. Wang, and J. Zhu, "Super-resolution digital holographic imaging method," Appl. Phys. Lett. 81, 3143-3145 (2002).
[CrossRef]

U. Schnars and W. P. O. Jüptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

2001

2000

1999

1993

1977

A. Jerri, "The Shannon sampling theorem - its various extensions and applications," Proc. IEEE 65, 1565-1596 (1977).
[CrossRef]

Alexandrov, S. A.

S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, "Synthetic aperture Fourier holographic optical microscopy," Phys. Rev. Lett. 97, 168102-1-4 (2006).
[CrossRef]

Alfieri, D.

L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, "Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram," Appl. Phys. Lett. 90, 041104-1-3 (2007).
[CrossRef]

Bo, F.

C. Liu, Z. Liu, F. Bo, Y. Wang, and J. Zhu, "Super-resolution digital holographic imaging method," Appl. Phys. Lett. 81, 3143-3145 (2002).
[CrossRef]

Collot, L.

Coppola, G.

P. Ferraro, C. Del Core, L. Miccio, S. Grilli, S. De Nicola, A. Finizio, and G. Coppola, "Phase map retrieval in digital holography: avoiding the undersampling effect by lateral shear approach," Opt. Lett. 32, 2233-2235 (2007).
[CrossRef] [PubMed]

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

de Groot, P.

De Nicola, D.

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

De Nicola, S.

De Petrocellis, L.

L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, "Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram," Appl. Phys. Lett. 90, 041104-1-3 (2007).
[CrossRef]

Deck, L.

Del Core, C.

Demoli, I.

N. Demoli and I. Demoli, "Dynamic modal characterization of musical instruments using digital holography," Opt. Express 13, 4812-4817 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-13-4812.
[CrossRef] [PubMed]

N. Demoli, and I. Demoli, "Measuring surface vibrations of musical instruments using an inexpensive digital holography device," Opt. Eng. 44, 0905502-1-3 (2005).
[CrossRef]

Demoli, N.

N. Demoli, M. Torzynski, and D. Vukicevic, "Enhanced sensitivity digital holographic interferometry," Opt. Express 15, 10672-10680 (2007).
[CrossRef] [PubMed]

N. Demoli, "Real-time monitoring of vibration fringe patterns by optical reconstructing of digital holograms: mode beating detection," Opt. Express 14, 2117-2122 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-6-2117.
[CrossRef] [PubMed]

N. Demoli and I. Demoli, "Dynamic modal characterization of musical instruments using digital holography," Opt. Express 13, 4812-4817 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-13-4812.
[CrossRef] [PubMed]

N. Demoli and D. Vukicevic, "Detection of hidden stationary deformations of vibrating surfaces by use of time-averaged digital holographic interferometry," Opt. Lett. 29, 1423-2425 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=ol-29-20-2423.
[CrossRef]

N. Demoli, D. Vukicevic, and M. Torzynski, "Time-averaged holographic interferometry using subtraction digital holography," Proc. SPIE 5457, 643-650 (2004).
[CrossRef]

N. Demoli, J. Meštrovi?, and I. Sovi?, "Subtraction digital holography," Appl. Opt. 42, 798-804 (2003).
[CrossRef] [PubMed]

N. Demoli, D. Vukicevic, and M. Torzynski, "Dynamic digital holographic interferometry with three wavelenths," Opt. Express 11, 767-774 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-7-767.
[CrossRef] [PubMed]

N. Demoli, and I. Demoli, "Measuring surface vibrations of musical instruments using an inexpensive digital holography device," Opt. Eng. 44, 0905502-1-3 (2005).
[CrossRef]

Desse, J. M.

Ferraro, P.

P. Ferraro, C. Del Core, L. Miccio, S. Grilli, S. De Nicola, A. Finizio, and G. Coppola, "Phase map retrieval in digital holography: avoiding the undersampling effect by lateral shear approach," Opt. Lett. 32, 2233-2235 (2007).
[CrossRef] [PubMed]

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

L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, "Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram," Appl. Phys. Lett. 90, 041104-1-3 (2007).
[CrossRef]

Finizio, A.

P. Ferraro, C. Del Core, L. Miccio, S. Grilli, S. De Nicola, A. Finizio, and G. Coppola, "Phase map retrieval in digital holography: avoiding the undersampling effect by lateral shear approach," Opt. Lett. 32, 2233-2235 (2007).
[CrossRef] [PubMed]

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

L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, "Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram," Appl. Phys. Lett. 90, 041104-1-3 (2007).
[CrossRef]

Garcia, J.

Garcia-Martinez, P.

Grilli, S.

P. Ferraro, C. Del Core, L. Miccio, S. Grilli, S. De Nicola, A. Finizio, and G. Coppola, "Phase map retrieval in digital holography: avoiding the undersampling effect by lateral shear approach," Opt. Lett. 32, 2233-2235 (2007).
[CrossRef] [PubMed]

L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, "Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram," Appl. Phys. Lett. 90, 041104-1-3 (2007).
[CrossRef]

Gross, M.

Gusev, M. E.

Gutzler, T.

S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, "Synthetic aperture Fourier holographic optical microscopy," Phys. Rev. Lett. 97, 168102-1-4 (2006).
[CrossRef]

Hayashi, Y.

Y. Takaki and Y. Hayashi, "Increased horizontal viewing zone angle of a hologram by resolution redistribution of a spatial light modulator," Appl. Opt. 47, 6-11 (2008).
[CrossRef]

Hillman, T. R.

S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, "Synthetic aperture Fourier holographic optical microscopy," Phys. Rev. Lett. 97, 168102-1-4 (2006).
[CrossRef]

Jerri, A.

A. Jerri, "The Shannon sampling theorem - its various extensions and applications," Proc. IEEE 65, 1565-1596 (1977).
[CrossRef]

Jüptner, W.

Jüptner, W. P. O.

U. Schnars and W. P. O. Jüptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

Kemper, B.

Kreis, T.

T. Kreis, "Frequency analysis of digital holography," Opt. Eng. 41, 771-778 (2002).
[CrossRef]

Le Clerc, F.

Liu, C.

C. Liu, Z. Liu, F. Bo, Y. Wang, and J. Zhu, "Super-resolution digital holographic imaging method," Appl. Phys. Lett. 81, 3143-3145 (2002).
[CrossRef]

Liu, Z.

C. Liu, Z. Liu, F. Bo, Y. Wang, and J. Zhu, "Super-resolution digital holographic imaging method," Appl. Phys. Lett. 81, 3143-3145 (2002).
[CrossRef]

Massig, J. H.

Meštrovic, J.

Miccio, L.

P. Ferraro, C. Del Core, L. Miccio, S. Grilli, S. De Nicola, A. Finizio, and G. Coppola, "Phase map retrieval in digital holography: avoiding the undersampling effect by lateral shear approach," Opt. Lett. 32, 2233-2235 (2007).
[CrossRef] [PubMed]

L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, "Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram," Appl. Phys. Lett. 90, 041104-1-3 (2007).
[CrossRef]

Mico, V.

Mishina, T.

Nicola, S. D.

L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, "Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram," Appl. Phys. Lett. 90, 041104-1-3 (2007).
[CrossRef]

Okano, F.

Okui, M.

Osten, W.

Pedrini, G.

Perez, P.

P. Perez and A. Santos, "Undersampling to acquire nuclear magnetic resonance images," Med. Eng. Phys. 26, 523-529 (2004).
[CrossRef] [PubMed]

Picart, P.

Pierattini, G.

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

Sampson, D. D.

S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, "Synthetic aperture Fourier holographic optical microscopy," Phys. Rev. Lett. 97, 168102-1-4 (2006).
[CrossRef]

Santos, A.

P. Perez and A. Santos, "Undersampling to acquire nuclear magnetic resonance images," Med. Eng. Phys. 26, 523-529 (2004).
[CrossRef] [PubMed]

Schnars, U.

U. Schnars and W. P. O. Jüptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

Seebacher, S.

Sovic, I.

Stadelmaier, A.

Takaki, Y.

Y. Takaki and Y. Hayashi, "Increased horizontal viewing zone angle of a hologram by resolution redistribution of a spatial light modulator," Appl. Opt. 47, 6-11 (2008).
[CrossRef]

Tankam, P.

Torzynski, M.

von Bally, G.

Vukicevic, D.

N. Demoli, M. Torzynski, and D. Vukicevic, "Enhanced sensitivity digital holographic interferometry," Opt. Express 15, 10672-10680 (2007).
[CrossRef] [PubMed]

N. Demoli, D. Vukicevic, and M. Torzynski, "Time-averaged holographic interferometry using subtraction digital holography," Proc. SPIE 5457, 643-650 (2004).
[CrossRef]

N. Demoli and D. Vukicevic, "Detection of hidden stationary deformations of vibrating surfaces by use of time-averaged digital holographic interferometry," Opt. Lett. 29, 1423-2425 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=ol-29-20-2423.
[CrossRef]

N. Demoli, D. Vukicevic, and M. Torzynski, "Dynamic digital holographic interferometry with three wavelenths," Opt. Express 11, 767-774 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-7-767.
[CrossRef] [PubMed]

Wagner, C.

Wang, Y.

C. Liu, Z. Liu, F. Bo, Y. Wang, and J. Zhu, "Super-resolution digital holographic imaging method," Appl. Phys. Lett. 81, 3143-3145 (2002).
[CrossRef]

Yuyama, I.

Zalevsky, Z.

Zhang, S.

S. Zhang, "Application of super-resolution image reconstruction to digital holography," EURASIP Journal on Applied Signal Processing 2006, Article ID 90358-1-7 (2006).

Zhu, J.

C. Liu, Z. Liu, F. Bo, Y. Wang, and J. Zhu, "Super-resolution digital holographic imaging method," Appl. Phys. Lett. 81, 3143-3145 (2002).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

C. Liu, Z. Liu, F. Bo, Y. Wang, and J. Zhu, "Super-resolution digital holographic imaging method," Appl. Phys. Lett. 81, 3143-3145 (2002).
[CrossRef]

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

J. Opt. Soc. Am. A

Meas. Sci. Technol.

U. Schnars and W. P. O. Jüptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

Med. Eng. Phys.

P. Perez and A. Santos, "Undersampling to acquire nuclear magnetic resonance images," Med. Eng. Phys. 26, 523-529 (2004).
[CrossRef] [PubMed]

Opt. Eng.

T. Kreis, "Frequency analysis of digital holography," Opt. Eng. 41, 771-778 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. IEEE

A. Jerri, "The Shannon sampling theorem - its various extensions and applications," Proc. IEEE 65, 1565-1596 (1977).
[CrossRef]

Proc. SPIE

N. Demoli, D. Vukicevic, and M. Torzynski, "Time-averaged holographic interferometry using subtraction digital holography," Proc. SPIE 5457, 643-650 (2004).
[CrossRef]

Other

N. Demoli, and I. Demoli, "Measuring surface vibrations of musical instruments using an inexpensive digital holography device," Opt. Eng. 44, 0905502-1-3 (2005).
[CrossRef]

D. A. Naylor, B. G. Gom, T. R. Fulton, M. K. Tahic, and G. R. Davis, "Increased efficiency through undersampling in Fourier transform spectroscopy," in Fourier Transform Spectroscopy/Hyperspectral Imaging and Sounding of the Environment, Technical Digest (CD) (Optical Society of America, 2005), paper FtuD14, http://www.opticsinfobase.org/abstract.cfm?URI=FTS-2005-FTuD14.
[PubMed]

S. Zhang, "Application of super-resolution image reconstruction to digital holography," EURASIP Journal on Applied Signal Processing 2006, Article ID 90358-1-7 (2006).

S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, "Synthetic aperture Fourier holographic optical microscopy," Phys. Rev. Lett. 97, 168102-1-4 (2006).
[CrossRef]

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

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 2000), Chap. 10.

E. T. Whittaker, Interpolary Function Theory, Cambridge Tracts in Mathematics and Mathematical Physics, 33 (Cambridge U. P., 1935).

C. E. Shannon, "Communication in the presence of noise," Proc. IRE 37, 10-21 (1949).
[CrossRef]

L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, "Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram," Appl. Phys. Lett. 90, 041104-1-3 (2007).
[CrossRef]

U. Schnars and W. Jüptner, Digital Holography (Springer-Verlag, 2005).

Supplementary Material (1)

» Media 1: AVI (7284 KB)     

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Figures (8)

Fig. 1.
Fig. 1.

Quasi-Fourier holography configuration. (P1) input plane; (P2) hologram plane.

Fig. 2.
Fig. 2.

Scheme of the experimental setup. (VA) variable attenuator; (M) mirrors; (BS) beam splitter; (SF) spatial filter; (ML) micro lens; (L1) cylindrical lens; (L2) lens; (Ob.) object.

Fig. 3.
Fig. 3.

Photograph of the membrane.

Fig. 4.
Fig. 4.

Effects of increasing lateral distance of the oscillating membrane (Media 1).

Fig. 5.
Fig. 5.

The reconstruction images demonstrating the 0th interval (upper row), 1th interval (middle row), and 2nd interval (lower row). The red arrow denotes the shift direction, while the minus sign denotes inversion.

Fig. 6.
Fig. 6.

Limits of the full interval defined by the experimental conditions.

Fig. 7.
Fig. 7.

Limits for the non-overlapping intervals obtained by introducing the object (a membrane of a diameter 32 mm) into analysis.

Fig. 8.
Fig. 8.

Same as in Fig. 7, but with the use of the subtraction method.

Equations (21)

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ψ (ξ)=R0δ(ξ)+s(ξb),
ψ (x)=ζ(x)FT{ψ(ξ)ζ(ξ)}=ζ(x)[R0+FT{s(ξb)ζ(ξ)}],
I (x)=DC(x)+R0FT{s(ξb)ζ(ξ)}+CC(x),
ψ (ξ)DC(ξ)+s(ξb)ζ*(b)+s(ξ+b)ζ(b),
ψ(ξ)DC(ξ)+s(ξb)+s(ξ+b).
I (x)=I0(x){1+V(x)cos[ϑ(x)+2πucx+φb]},
Ic (x)=I(x)rect(xL),
ψc (ξ)=[DC(ξ)+s(ξb)ζ*(b)+s(ξ+b)ζ(b)]sinc[L(λd)1ξ],
Id (x)=[I(x)rect(xαΔx)]comb(xΔx)rect(xNΔx),
Id (x)=I0(x){1+V(x)cos[ϑ(x)+2πucx+φb]}comb(xΔx)rect(xNΔx).
ψd (kΔξ) [DC(kΔξ+s(kΔξkΔbΔξ)ζ*(kΔbΔξ)+s(kΔξ+kΔbΔξ)ζ(kΔbΔξ)]
sinc[NΔx(λd)1kΔξ]comb[Δx(λd)1kΔξ],
[ius2+12(WDC+Wo),(i+1)us2Wo2],i=0,2,4,(even)
[ius2+Wo2,(i+1)us212(WDC+Wo)],i=1,3,5,(odd)
[tan1(N+3B2d),tan1((i+1)ξNB2d)] , i = 0 , 2 , 4 , (even)
[tan1(N+B2d),tan1((i+1)ξN3B2d)] , i = 1 , 3 , 5 , (odd)
[ius2+12Wo,(i+1)us2Wo2],i=0,2,4,(even)
[ius2+Wo2,(i+1)us2Wo2],i=1,3,5,(odd)
[tan1(N+B2d),tan1((i+1)ξNB2d)] , i = 0 , 2 , 4 , (even)
[tan1(N+B2d),tan1((i+1)ξNB2d)] , i = 1 , 3 , 5 , (odd)
ϕ = λd2ΔxB ,

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