Abstract

In this work, we present a new method to reduce the shot noise in phase imaging of digital holograms. A spatial averaging process of phase images reconstructed at different reconstruction distances is performed, with the reconstruction distance range being specified by the numerical focus depth of the optical system. An improved phase image is attained with a 50% shot noise reduction. We use the integral of the angular spectrum as a reconstruction method to obtain a single-object complex amplitude that is needed to perform our proposal. We also show the corresponding simulations and experimental results. The topography of a homemade TiO2 stepwise of 100 nm high was measured and compared with the atomic force microscope results.

© 2012 Optical Society of America

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  1. J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
    [CrossRef]
  2. M. A. Kronrod, N. S. Merzlyakov, and P. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 333–334 (1972).
  3. U. Schnars and W. Jüptner, “Direct recording of holograms by to CCD-target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994).
    [CrossRef]
  4. V. Kebbel, H. J. Hartmann, and W. Jüptner, “New approach for testing of aspherical micro-optics with high numerical aperture,” Proc. SPIE 4451, 345–355 (2001).
    [CrossRef]
  5. C. Furlong and R. J. Pryputniewics, “Optoelectronic characterization of shape and deformations of MEMS accelerometers used in transportation applications,” Opt. Eng. 42, 1223–1231 (2003).
    [CrossRef]
  6. F. Charrière, J. Kühn, T. Colomb, F. Monfort, E. Cuche, Y. Emery, K. Weible, P. Marquet, and C. Depeursinge, “Characterization of microlenses by digital holographic microscopy,” Appl. Opt. 45, 829–835 (2006).
    [CrossRef]
  7. B. Kemper, S. Stürwald, C. Remmersmann, P. Langehanenbergerg, and G. Von Bally, “Characterization of light emitting diodes (LEDs) for applications in digital holographic microscopy for inspection of micro and nanostructured surfaces,” Opt. Eng. 46, 499–507 (2008).
    [CrossRef]
  8. D. Carl, B. Kemper, G. Wernicke, and G. Von Bally, “Parameter optimized digital holographic microscope for high resolution living cells analysis,” Appl. Opt. 43, 6536–6544 (2004).
    [CrossRef]
  9. H. Sun, M. Player, J. Watson, D. Hendry, R. Perkins, G. Gust, and D. Paterson, “The use of digital/electronic holography for biological applications,” J. Opt. A 7, S399–S407 (2005).
    [CrossRef]
  10. B. Rappaz, P. Marquet, E. Cuche, Y. Emery, C. Depeursinge, and P. Magistretti, “Measurement of the integral refractive index and dynamic cell morphometry of living cells with digital holographic microscopy,” Opt. Express 13, 9361–9373 (2005).
    [CrossRef]
  11. C. Mann, L. Yu, and M. Kim, “Movies of cellular and sub-cellular motion by digital holographic microscopy,” Biomed. Eng. 5, 21–31 (2006).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  22. J. kühn, F. Charrière, T. Colomb, E. Cuche, F. Montfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
    [CrossRef]
  23. M. Potcoava and M. Km, “Fingerprint biometry applications of digital holography and low-coherence interferography,” Appl. Opt. 48, H9–H15 (2009).
    [CrossRef]
  24. F. Dubois, L. Joannes, and J. Legros, “Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence,” Appl. Opt. 38, 7085–7094 (1999).
    [CrossRef]
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    [CrossRef]
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  28. A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 2002).

2010 (1)

2009 (2)

M. Potcoava and M. Km, “Fingerprint biometry applications of digital holography and low-coherence interferography,” Appl. Opt. 48, H9–H15 (2009).
[CrossRef]

J. I. Di, J. Zhao, W. Sun, H. Jiang, and X. Yan, “Phase aberration compensation of digital holographic microscopy based on least squares surface fitting,” Opt. Commun. 282, 3873–3877 (2009).
[CrossRef]

2008 (4)

J. Garcia-Sucerquia, W. Xu, S. K. Jericho, M. H. Jericho, and H. J. Kreuzer, “4-D imaging of fluid flow digital with digital in-line holographic microscopy,” Optik 119, 419–423 (2008).
[CrossRef]

B. Kemper, S. Stürwald, C. Remmersmann, P. Langehanenbergerg, and G. Von Bally, “Characterization of light emitting diodes (LEDs) for applications in digital holographic microscopy for inspection of micro and nanostructured surfaces,” Opt. Eng. 46, 499–507 (2008).
[CrossRef]

J. kühn, F. Charrière, T. Colomb, E. Cuche, F. Montfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
[CrossRef]

X. Kang, “An effective method for reducing speckle noise in digital holography,” Chin. Opt. Lett. 6, 100–103 (2008).
[CrossRef]

2007 (1)

2006 (5)

2005 (3)

2004 (2)

2003 (1)

C. Furlong and R. J. Pryputniewics, “Optoelectronic characterization of shape and deformations of MEMS accelerometers used in transportation applications,” Opt. Eng. 42, 1223–1231 (2003).
[CrossRef]

2001 (1)

V. Kebbel, H. J. Hartmann, and W. Jüptner, “New approach for testing of aspherical micro-optics with high numerical aperture,” Proc. SPIE 4451, 345–355 (2001).
[CrossRef]

1999 (1)

1997 (1)

1994 (1)

1972 (1)

M. A. Kronrod, N. S. Merzlyakov, and P. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 333–334 (1972).

1967 (1)

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
[CrossRef]

Alferi, D.

Aspert, N.

Baumbach, T.

Bergmann, L.

L. Bergmann and C. Schaefer, Optics of Waves and Particles (Gruyter, 2003).

Carl, D.

Charrière, F.

Colenovic, E.

Colomb, T.

Coppola, G.

Cuche, E.

Depeursinge, C.

Di, J. I.

J. I. Di, J. Zhao, W. Sun, H. Jiang, and X. Yan, “Phase aberration compensation of digital holographic microscopy based on least squares surface fitting,” Opt. Commun. 282, 3873–3877 (2009).
[CrossRef]

Dubois, F.

Emery, Y.

Ferraro, P.

Finizio, A.

Furlong, C.

C. Furlong and R. J. Pryputniewics, “Optoelectronic characterization of shape and deformations of MEMS accelerometers used in transportation applications,” Opt. Eng. 42, 1223–1231 (2003).
[CrossRef]

Garcia-Sucerquia, J.

J. Garcia-Sucerquia, W. Xu, S. K. Jericho, M. H. Jericho, and H. J. Kreuzer, “4-D imaging of fluid flow digital with digital in-line holographic microscopy,” Optik 119, 419–423 (2008).
[CrossRef]

Goodman, J. W.

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
[CrossRef]

J. W. Goodman, Statistical Optics (Wiley, 1985).

Grilli, S.

Gust, G.

H. Sun, M. Player, J. Watson, D. Hendry, R. Perkins, G. Gust, and D. Paterson, “The use of digital/electronic holography for biological applications,” J. Opt. A 7, S399–S407 (2005).
[CrossRef]

Hartmann, H. J.

V. Kebbel, H. J. Hartmann, and W. Jüptner, “New approach for testing of aspherical micro-optics with high numerical aperture,” Proc. SPIE 4451, 345–355 (2001).
[CrossRef]

Hendry, D.

H. Sun, M. Player, J. Watson, D. Hendry, R. Perkins, G. Gust, and D. Paterson, “The use of digital/electronic holography for biological applications,” J. Opt. A 7, S399–S407 (2005).
[CrossRef]

Istasse, E.

Javidi, B.

Jericho, M. H.

J. Garcia-Sucerquia, W. Xu, S. K. Jericho, M. H. Jericho, and H. J. Kreuzer, “4-D imaging of fluid flow digital with digital in-line holographic microscopy,” Optik 119, 419–423 (2008).
[CrossRef]

Jericho, S. K.

J. Garcia-Sucerquia, W. Xu, S. K. Jericho, M. H. Jericho, and H. J. Kreuzer, “4-D imaging of fluid flow digital with digital in-line holographic microscopy,” Optik 119, 419–423 (2008).
[CrossRef]

Jiang, H.

J. I. Di, J. Zhao, W. Sun, H. Jiang, and X. Yan, “Phase aberration compensation of digital holographic microscopy based on least squares surface fitting,” Opt. Commun. 282, 3873–3877 (2009).
[CrossRef]

Joannes, L.

Jüptner, W.

Kang, X.

Kebbel, V.

T. Baumbach, E. Colenovic, V. Kebbel, and W. Jüptner, “Improvement of accuracy in digital holography by uses of multiple holograms,” Appl. Opt. 45, 6077–6085 (2006).
[CrossRef]

V. Kebbel, H. J. Hartmann, and W. Jüptner, “New approach for testing of aspherical micro-optics with high numerical aperture,” Proc. SPIE 4451, 345–355 (2001).
[CrossRef]

Kemper, B.

B. Kemper, S. Stürwald, C. Remmersmann, P. Langehanenbergerg, and G. Von Bally, “Characterization of light emitting diodes (LEDs) for applications in digital holographic microscopy for inspection of micro and nanostructured surfaces,” Opt. Eng. 46, 499–507 (2008).
[CrossRef]

D. Carl, B. Kemper, G. Wernicke, and G. Von Bally, “Parameter optimized digital holographic microscope for high resolution living cells analysis,” Appl. Opt. 43, 6536–6544 (2004).
[CrossRef]

Kim, M.

C. Mann, L. Yu, and M. Kim, “Movies of cellular and sub-cellular motion by digital holographic microscopy,” Biomed. Eng. 5, 21–31 (2006).

Km, M.

Kreuzer, H. J.

J. Garcia-Sucerquia, W. Xu, S. K. Jericho, M. H. Jericho, and H. J. Kreuzer, “4-D imaging of fluid flow digital with digital in-line holographic microscopy,” Optik 119, 419–423 (2008).
[CrossRef]

Kronrod, M. A.

M. A. Kronrod, N. S. Merzlyakov, and P. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 333–334 (1972).

kühn, J.

Langehanenbergerg, P.

B. Kemper, S. Stürwald, C. Remmersmann, P. Langehanenbergerg, and G. Von Bally, “Characterization of light emitting diodes (LEDs) for applications in digital holographic microscopy for inspection of micro and nanostructured surfaces,” Opt. Eng. 46, 499–507 (2008).
[CrossRef]

Lawrence, R. W.

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
[CrossRef]

Legros, J.

Li, R.

Liu, S.

Magistretti, P.

Mann, C.

C. Mann, L. Yu, and M. Kim, “Movies of cellular and sub-cellular motion by digital holographic microscopy,” Biomed. Eng. 5, 21–31 (2006).

Market, P.

Marquet, P.

Merzlyakov, N. S.

M. A. Kronrod, N. S. Merzlyakov, and P. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 333–334 (1972).

Minetti, C.

Monfort, F.

Monnom, O.

Montfort, F.

J. kühn, F. Charrière, T. Colomb, E. Cuche, F. Montfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
[CrossRef]

Nicola, S. Of

Novella, M.

Pan, F.

Papoulis, A.

A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 2002).

Paterson, D.

H. Sun, M. Player, J. Watson, D. Hendry, R. Perkins, G. Gust, and D. Paterson, “The use of digital/electronic holography for biological applications,” J. Opt. A 7, S399–S407 (2005).
[CrossRef]

Perkins, R.

H. Sun, M. Player, J. Watson, D. Hendry, R. Perkins, G. Gust, and D. Paterson, “The use of digital/electronic holography for biological applications,” J. Opt. A 7, S399–S407 (2005).
[CrossRef]

Pieranttini, G.

Pillai, S. U.

A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 2002).

Player, M.

H. Sun, M. Player, J. Watson, D. Hendry, R. Perkins, G. Gust, and D. Paterson, “The use of digital/electronic holography for biological applications,” J. Opt. A 7, S399–S407 (2005).
[CrossRef]

Potcoava, M.

Pryputniewics, R. J.

C. Furlong and R. J. Pryputniewics, “Optoelectronic characterization of shape and deformations of MEMS accelerometers used in transportation applications,” Opt. Eng. 42, 1223–1231 (2003).
[CrossRef]

Rappaz, B.

Remmersmann, C.

B. Kemper, S. Stürwald, C. Remmersmann, P. Langehanenbergerg, and G. Von Bally, “Characterization of light emitting diodes (LEDs) for applications in digital holographic microscopy for inspection of micro and nanostructured surfaces,” Opt. Eng. 46, 499–507 (2008).
[CrossRef]

Rong, L.

Schaefer, C.

L. Bergmann and C. Schaefer, Optics of Waves and Particles (Gruyter, 2003).

Schnars, U.

Striano, V.

Stürwald, S.

B. Kemper, S. Stürwald, C. Remmersmann, P. Langehanenbergerg, and G. Von Bally, “Characterization of light emitting diodes (LEDs) for applications in digital holographic microscopy for inspection of micro and nanostructured surfaces,” Opt. Eng. 46, 499–507 (2008).
[CrossRef]

Sun, H.

H. Sun, M. Player, J. Watson, D. Hendry, R. Perkins, G. Gust, and D. Paterson, “The use of digital/electronic holography for biological applications,” J. Opt. A 7, S399–S407 (2005).
[CrossRef]

Sun, W.

J. I. Di, J. Zhao, W. Sun, H. Jiang, and X. Yan, “Phase aberration compensation of digital holographic microscopy based on least squares surface fitting,” Opt. Commun. 282, 3873–3877 (2009).
[CrossRef]

Von Bally, G.

B. Kemper, S. Stürwald, C. Remmersmann, P. Langehanenbergerg, and G. Von Bally, “Characterization of light emitting diodes (LEDs) for applications in digital holographic microscopy for inspection of micro and nanostructured surfaces,” Opt. Eng. 46, 499–507 (2008).
[CrossRef]

D. Carl, B. Kemper, G. Wernicke, and G. Von Bally, “Parameter optimized digital holographic microscope for high resolution living cells analysis,” Appl. Opt. 43, 6536–6544 (2004).
[CrossRef]

Watson, J.

H. Sun, M. Player, J. Watson, D. Hendry, R. Perkins, G. Gust, and D. Paterson, “The use of digital/electronic holography for biological applications,” J. Opt. A 7, S399–S407 (2005).
[CrossRef]

Weible, K.

Wernicke, G.

Xiao, W.

Xu, W.

J. Garcia-Sucerquia, W. Xu, S. K. Jericho, M. H. Jericho, and H. J. Kreuzer, “4-D imaging of fluid flow digital with digital in-line holographic microscopy,” Optik 119, 419–423 (2008).
[CrossRef]

Yamaguchi, I.

Yan, X.

J. I. Di, J. Zhao, W. Sun, H. Jiang, and X. Yan, “Phase aberration compensation of digital holographic microscopy based on least squares surface fitting,” Opt. Commun. 282, 3873–3877 (2009).
[CrossRef]

Yaroslavskii, P.

M. A. Kronrod, N. S. Merzlyakov, and P. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 333–334 (1972).

Yu, L.

C. Mann, L. Yu, and M. Kim, “Movies of cellular and sub-cellular motion by digital holographic microscopy,” Biomed. Eng. 5, 21–31 (2006).

Zhang, T.

Zhao, J.

J. I. Di, J. Zhao, W. Sun, H. Jiang, and X. Yan, “Phase aberration compensation of digital holographic microscopy based on least squares surface fitting,” Opt. Commun. 282, 3873–3877 (2009).
[CrossRef]

Appl. Opt. (8)

U. Schnars and W. Jüptner, “Direct recording of holograms by to CCD-target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994).
[CrossRef]

F. Charrière, J. Kühn, T. Colomb, F. Monfort, E. Cuche, Y. Emery, K. Weible, P. Marquet, and C. Depeursinge, “Characterization of microlenses by digital holographic microscopy,” Appl. Opt. 45, 829–835 (2006).
[CrossRef]

D. Carl, B. Kemper, G. Wernicke, and G. Von Bally, “Parameter optimized digital holographic microscope for high resolution living cells analysis,” Appl. Opt. 43, 6536–6544 (2004).
[CrossRef]

T. Colomb, E. Cuche, F. Charrière, J. Kühn, N. Aspert, F. Monfort, P. Marquet, and C. Depeursinge, “Automatic procedure for aberration compensation in digital holographic microscopy and applications to specimen shape compensation,” Appl. Opt. 45, 851–863 (2006).
[CrossRef]

T. Baumbach, E. Colenovic, V. Kebbel, and W. Jüptner, “Improvement of accuracy in digital holography by uses of multiple holograms,” Appl. Opt. 45, 6077–6085 (2006).
[CrossRef]

F. Dubois, M. Novella, C. Minetti, O. Monnom, and E. Istasse, “Partial spatial coherence effects in digital holographic microscopy with to laser source,” Appl. Opt. 43, 1131–1139 (2004).
[CrossRef]

M. Potcoava and M. Km, “Fingerprint biometry applications of digital holography and low-coherence interferography,” Appl. Opt. 48, H9–H15 (2009).
[CrossRef]

F. Dubois, L. Joannes, and J. Legros, “Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence,” Appl. Opt. 38, 7085–7094 (1999).
[CrossRef]

Appl. Phys. Lett. (1)

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
[CrossRef]

Biomed. Eng. (1)

C. Mann, L. Yu, and M. Kim, “Movies of cellular and sub-cellular motion by digital holographic microscopy,” Biomed. Eng. 5, 21–31 (2006).

Chin. Opt. Lett. (2)

J. Opt. A (1)

H. Sun, M. Player, J. Watson, D. Hendry, R. Perkins, G. Gust, and D. Paterson, “The use of digital/electronic holography for biological applications,” J. Opt. A 7, S399–S407 (2005).
[CrossRef]

Meas. Sci. Technol. (1)

J. kühn, F. Charrière, T. Colomb, E. Cuche, F. Montfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
[CrossRef]

Opt. Commun. (1)

J. I. Di, J. Zhao, W. Sun, H. Jiang, and X. Yan, “Phase aberration compensation of digital holographic microscopy based on least squares surface fitting,” Opt. Commun. 282, 3873–3877 (2009).
[CrossRef]

Opt. Eng. (2)

C. Furlong and R. J. Pryputniewics, “Optoelectronic characterization of shape and deformations of MEMS accelerometers used in transportation applications,” Opt. Eng. 42, 1223–1231 (2003).
[CrossRef]

B. Kemper, S. Stürwald, C. Remmersmann, P. Langehanenbergerg, and G. Von Bally, “Characterization of light emitting diodes (LEDs) for applications in digital holographic microscopy for inspection of micro and nanostructured surfaces,” Opt. Eng. 46, 499–507 (2008).
[CrossRef]

Opt. Express (4)

Opt. Lett. (1)

Optik (1)

J. Garcia-Sucerquia, W. Xu, S. K. Jericho, M. H. Jericho, and H. J. Kreuzer, “4-D imaging of fluid flow digital with digital in-line holographic microscopy,” Optik 119, 419–423 (2008).
[CrossRef]

Proc. SPIE (1)

V. Kebbel, H. J. Hartmann, and W. Jüptner, “New approach for testing of aspherical micro-optics with high numerical aperture,” Proc. SPIE 4451, 345–355 (2001).
[CrossRef]

Sov. Phys. Tech. Phys. (1)

M. A. Kronrod, N. S. Merzlyakov, and P. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 333–334 (1972).

Other (3)

J. W. Goodman, Statistical Optics (Wiley, 1985).

A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 2002).

L. Bergmann and C. Schaefer, Optics of Waves and Particles (Gruyter, 2003).

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Figures (16)

Fig. 1.
Fig. 1.

Normalized spectrum distribution of the light intensity emitted by the commercial ultrabrilliant LED.

Fig. 2.
Fig. 2.

Schematic of the microscope implemented on a Mach–Zehnder interferometer with a spatially partial coherent source for digital holographic microscopy. Abbreviations are defined in text.

Fig. 3.
Fig. 3.

STD of the reconstructed images using the AS reconstruction method as a function of the optical power.

Fig. 4.
Fig. 4.

Correlation coefficients between reconstructed phase images at different reconstruction distances from the same complex amplitude.

Fig. 5.
Fig. 5.

Effect of averaging on a series of simulated holograms for a mean number of photons per pixel of 100, 500, 4000, and 8000: STD of the performed phase image as a function of the number of phase images C used in the averaging procedure.

Fig. 6.
Fig. 6.

Correlation coefficients between reconstructed phase images at different reconstruction distances from the same complex amplitude with a magnification of 20.

Fig. 7.
Fig. 7.

Reconstructed phase image from an experimentally recorded hologram without any specimen in the system. (a) Phase image without aberration correction where DST=12.6deg is measured, delimited by the black square. (b) Phase image with aberration correction using the RCH method, where DST=0.7deg is measured, delimited by the black square.

Fig. 8.
Fig. 8.

Correlation coefficient between reconstructed phase images at different reconstruction distances from the same complex amplitude, which comes from real holograms with an average number of photons per pixel of 5100.

Fig. 9.
Fig. 9.

Shot noise reduction through proposed averaging technique of experimental holograms. (a) Single reconstructed phase image with a phase STD of 0.693 deg. (b) Ten averaged phase images reconstructed with a Δd=2μm to each other with a phase DST of 0.231 deg.

Fig. 10.
Fig. 10.

Effect of phase DST reduction occurs when the proposed averaging method is applied. (a) Measurement of phase DST of the images in Fig. 9, where the profile is indicated by a white line. One can clearly see a diminution of DST when 10 phase images are averaged. (b) Measurement of phase DST as a function of the number of phase images, where a behavior of C1/2 is shown in the DST reduction.

Fig. 11.
Fig. 11.

Illustration of the experimentally determined DOF. (a) One of the recorded intensity holograms from an Edmund NBS 1963A resolution card and the profile taken to measure the DOF. (b) Evolution of intensity determined on a profile line when reconstruction distance is increased.

Fig. 12.
Fig. 12.

Reconstructed focused amplitude images. (a) Reconstructed focused amplitude image without averaging. (b) Reconstructed amplitude image when the averaging process is performed with four focused amplitude images.

Fig. 13.
Fig. 13.

Comparison between profiles measured along the white lines defined in Figs. 12(a), 12(b).

Fig. 14.
Fig. 14.

Topographic measurement of the TiO2 stepwise specimen. (a) One reconstructed phase image. (b) Improved phase image when the averaging process is performed. (c) Corresponding zoomed area inside the white dashed rectangle in (b). (d) Numerical data extracted from AFM.

Fig. 15.
Fig. 15.

Comparison between profiles measured along the white lines defined on Figs. 14(c), and 14(d).

Fig. 16.
Fig. 16.

3D TiO2 step images. (a) Topographic measurement done by AFM, (b) topographic measurement enhanced through the averaging process done by DHM.

Equations (10)

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I(x,y)=|O(x,y)+R(x,y)|2=O2(x,y)+R2(x,y)+O(x,y)R*(x,y)+O*(x,y)R(x,y),
ϕ0(x,y)=tan1[(I4(x,y;3π/2)I2(x,y;π/2))(I1(x,y;0)I3(x,y;π))].
Ua(x,y)=A0(x,y)exp[i(ϕ0(x,y)+ϕ(x,y))],
U0(x,y)=A0(x,y)×exp[i(ϕ0(x,y)+ϕ(x,y)ϕ(x,y))].
U(x,y)=I1{exp[ikd(1αλβλ)1/2]×[IU0(x,y)](α,β)}(x,y),
U(mΔξ,nΔη)=FFT1{exp[ikd(1λr2λs2)]×FFT[U0(k,l)](r,s)}(m,n),
DOF=ΔξM2NA,
h=[λ(Δϕ/2π)/(nn0)],
σx¯=1CSTDc,
CCp,q=mNnN[Ip(m,n)Ip][Iq(m,n)Iq][mNnN[Ip(m,n)Ip]2mNnN[Iq(m,n)Iq]2]1/2,

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