Abstract

In this paper, we have investigated on the potentialities of digital holography for whole reconstruction of wavefields. We show that this technique can be efficiently used for obtaining quantitative information from the intensity and the phase distributions of the reconstructed field at different locations along the propagation direction. The basic concept and procedure of wavefield reconstruction for digital in-line holography is discussed. Numerical reconstructions of the wavefield from digitally recorded in-line hologram patterns and from simulated test patterns are presented. The potential of the method for analysing aberrated wave front has been exploited by applying the reconstruction procedure to astigmatic hologram patterns.

© Optical Society of America

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References

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  1. E. Cuche, P. Marquet, and C. Depeursinge, �Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,� Appl. Opt. 38, 6994-7001 (1999).
    [CrossRef]
  2. Y. Takaki and H. Ohzu, �Fast numerical reconstruction technique for high-resolution hybrid holographic microscopy,� Appl. Opt. 38, 2204-2211 (1999).
    [CrossRef]
  3. G. Pedrini, P. Fr�ning, H. Tiziani, F. Santoyo, �Shape measurement of microscopic structures using digital holograms,� Opt. Commun. 164, 257-268 (1999).
    [CrossRef]
  4. S. Schedin, G. Pedrini, H. Tiziani, A. K. Aggarwal, and M. E. Gusev, �Highly sensitive pulsed digital holography for built-in defect analysis with a laser excitation,� Appl. Opt. 40, 100-117 (2001).
    [CrossRef]
  5. T. Kreis, M. Adams, W. J�ptner, �Digital in-line holography in particle measurement,� SPIE 3744, 54-64 (1999).
    [CrossRef]
  6. S. Murata, N. Yasuda, �Potential of digital holography in particle measurement,� Optics & Laser Technology 32, 567-574 (2000).
    [CrossRef]
  7. J.W. Goodman, Introduction to Fourier Optics, McGraw-Hill, San Francisco, Calif., 1968, Cap. 5.
  8. Shinya Inoue and Kenneth R. Spring, Video Microscopy, Second Edition, Cap.7.
  9. S. De Nicola, P. Ferraro, A. Finizio and G. Pierattini, �Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase-shifting digital holography,� to be published, Opt. Las. Eng. (2001).
  10. A. Stadelmaier and J. H. Massing, �Compensation of lens aberrations in digital holography,� Opt. Lett. 25, 1630- 1632 (2000).
    [CrossRef]
  11. S. De Nicola, P. Ferraro, A. Finizio and G. Pierattini, �Correct-image reconstruction in the presence of severe anamorphism by means of digital holography,� Opt. Lett. 26, No.13, 974-976 (2001).
    [CrossRef]
  12. T.M. Kreis, W. P. O. J�ptner, Trends in Optical Non-Destructive testing and Inspection, Editors Pramod Rastogi and Daniele Inaudi, 113-127.
  13. Songcan Lai, Brian King, MArk A. Neifeld, �Wave front reconstruction by means of phase-shifting digital in-line holography,� Opt. Commun. 173, 155-160 (2000).
    [CrossRef]
  14. Takeda, Ina H. and Kobayashys, "Fourier transform method of fringe pattern analysis for computer based topography and interferometry," J. Opt. Soc. Am., 72, 156-160 (1982).
    [CrossRef]

Other

E. Cuche, P. Marquet, and C. Depeursinge, �Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,� Appl. Opt. 38, 6994-7001 (1999).
[CrossRef]

Y. Takaki and H. Ohzu, �Fast numerical reconstruction technique for high-resolution hybrid holographic microscopy,� Appl. Opt. 38, 2204-2211 (1999).
[CrossRef]

G. Pedrini, P. Fr�ning, H. Tiziani, F. Santoyo, �Shape measurement of microscopic structures using digital holograms,� Opt. Commun. 164, 257-268 (1999).
[CrossRef]

S. Schedin, G. Pedrini, H. Tiziani, A. K. Aggarwal, and M. E. Gusev, �Highly sensitive pulsed digital holography for built-in defect analysis with a laser excitation,� Appl. Opt. 40, 100-117 (2001).
[CrossRef]

T. Kreis, M. Adams, W. J�ptner, �Digital in-line holography in particle measurement,� SPIE 3744, 54-64 (1999).
[CrossRef]

S. Murata, N. Yasuda, �Potential of digital holography in particle measurement,� Optics & Laser Technology 32, 567-574 (2000).
[CrossRef]

J.W. Goodman, Introduction to Fourier Optics, McGraw-Hill, San Francisco, Calif., 1968, Cap. 5.

Shinya Inoue and Kenneth R. Spring, Video Microscopy, Second Edition, Cap.7.

S. De Nicola, P. Ferraro, A. Finizio and G. Pierattini, �Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase-shifting digital holography,� to be published, Opt. Las. Eng. (2001).

A. Stadelmaier and J. H. Massing, �Compensation of lens aberrations in digital holography,� Opt. Lett. 25, 1630- 1632 (2000).
[CrossRef]

S. De Nicola, P. Ferraro, A. Finizio and G. Pierattini, �Correct-image reconstruction in the presence of severe anamorphism by means of digital holography,� Opt. Lett. 26, No.13, 974-976 (2001).
[CrossRef]

T.M. Kreis, W. P. O. J�ptner, Trends in Optical Non-Destructive testing and Inspection, Editors Pramod Rastogi and Daniele Inaudi, 113-127.

Songcan Lai, Brian King, MArk A. Neifeld, �Wave front reconstruction by means of phase-shifting digital in-line holography,� Opt. Commun. 173, 155-160 (2000).
[CrossRef]

Takeda, Ina H. and Kobayashys, "Fourier transform method of fringe pattern analysis for computer based topography and interferometry," J. Opt. Soc. Am., 72, 156-160 (1982).
[CrossRef]

Supplementary Material (3)

» Media 1: MOV (873 KB)     
» Media 2: MOV (1716 KB)     
» Media 3: MOV (1716 KB)     

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Figures (9)

Fig. 1.
Fig. 1.

Experimental setup of the Mach-Zehnder interferometer for digital in-line holography.

Fig. 2.
Fig. 2.

Hologram recorded under two different conditions corresponding to two settings of the CCD frame buffer: (a) (736×572); (b) (768×572).

Fig. 3.
Fig. 3.

Clip video presentation of the digitally reconstructed object wavefield intensity distribution obtained at different distances d’ from the hologram plane along the z-axis direction, through the evaluation of the diffraction formula (1). The left video (873 KB) is obtained using the hologram pattern in Fig. 2a and for values of d’ ranging from 170 mm to 200 mm with discrete spatial step of Δz=1 mm; the right movie (1.903 KB) is obtained from the aberrated hologram of Fig. 2b and for d’ values ranging from 181 mm to 218 mm, with Δz=1 mm. Click on the figure with mouse to see the movie (<2 Mb for each). [Media 1] [Media 2] [Media 3]

Fig. 4.
Fig. 4.

Recording geometry in digital holographic reconstruction of the object wavefield.

Fig. 5.
Fig. 5.

Intensity distributions the circular (a) and elliptical (b) fringe patterns computed by Eq.(6)

Fig. 6.
Fig. 6.

Digital intensity reconstruction of the simulated hologram patterns of fig. 5a-5b: (a) reconstruction of the spherical wave front at distance d’=180 mm from the hologram plane; (b) image reconstruction of the astigmatic fringe pattern at a distance d’=180 mm (c) reconstruction of the spherical wave front at the focal plane z=250 mm; (d) reconstructed tangential focal line for the astigmatic fringe pattern at distance zy =250 mm

Fig. 7.
Fig. 7.

Wrapped phase distributions computed by the convolution-based reconstruction method at distance d’=180 mm (a) phase reconstruction of the simulated spherical wave front; (b) phase reconstruction of the astigmatic wave front.

Fig. 8.
Fig. 8.

Three-dimensional representations of the unwrapped phase values from the wrapped data of fig. 7a and 7b.

Fig. 9.
Fig. 9.

One dimensional representation of the unwrapped phase values along the x-horizontal (straight line) and y-vertical (dashed line) directions for reconstruction distances d’ ranging from 160 mm to 220 mm, step size of 10 mm: (a) phase reconstructions of the simulated spherical wave front ; (b) phase reconstructions of the astigmatic wave front. The scale of the horizontal axis is determined by the pixel size Δx’ξ of the reconstructed image which does not change in the convolution-based reconstruction method. The vertical axis is the z propagation axis along which the various phase distributions are evaluated.

Equations (12)

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b ( x , y ; d ) = exp { i π d λ ( ν 2 + μ 2 ) } h ( ξ , η ) r ( ξ , η ) g ( ξ , η )
× exp { 2 i π ( ξ ν + η μ ) } d ξ d η
g ( ξ , η ) = exp ( i 2 π d λ ) i λ d exp { i π d λ ( ξ 2 + η 2 ) }
Δ x = d λ N Δ ξ Δ y = d λ N Δ η
b ( x , y ; d ) = 1 { [ h ( ξ , η ) r ( ξ , η ) ] [ g ( ξ , η ) ] }
[ g ( ξ , η ) ] = g ( ξ , η ) exp [ i 2 π ( ν ξ + μ η ) ] d ξ d η
[ g ( ξ , η ) ] = exp ( i 2 π d / λ ) exp i π λ d ( ν 2 + μ 2 )
I ( ξ , η ) = l + cos [ π λ ( ξ 2 z x + η 2 z y ) ]
I ( ξ , η ) = 1 + 1 2 exp [ + i π λ ( ξ 2 z x + η 2 z y ) ] + 1 2 exp [ i π λ ( ξ 2 z x + η 2 z y ) ]
h ( ξ , η ) = 1 2 exp [ + i ψ ( ξ , η ) ]
ψ ( ξ , η ) = π λ ( ξ 2 z x + η 2 z y )
Δ ψ ( x , y , Δ z ) = Arctan [ Re { b ( x , y , d ) } Im { b ( x , y , d + Δ z ) } Re { b ( x , y , d + Δ z ) } Im { b ( x , y , d ) } Re { b ( x , y , d ) } Re { b ( x , y , d + Δ z ) } + Im { b ( x , y , d + Δ z ) } Im { b ( x , y , d ) } ]

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