Journal of the European Optical Society - Rapid publications, Vol 5 (2010)

General formulation of digital in-line holography from correlation with a chirplet function

S. Coëtmellec, N. Verrier, M. Brunel, D. Lebrun


Digital in-line holography is revisited to propose a mathematical model that describes the recording-reconstruction process as a linear shift-invariant system with a pseudo-point spread function even when the images are out of the optimal plane in the sense of signal processing. A particular case is treated to show that the optimal plane is the best focus plane in the sense of optics. Next, an exact solution of the holographic reconstruction by correlation is given. By means of the previous results, we study the behavior of the result of the correlation function between the diffraction pattern function produced by an opaque disk and a chirplet function and between the diffraction pattern produced by a phase disk and the same chirplet function.

© The Authors. All rights reserved. [DOI: 10.2971/jeos.2010.10027]

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S. Mann, and S. Haykin, "The chirplet transform: physical considerations" IEEE T. Signal Proces. 43, 2745-2761 (1995).

M. Born, and E. Wolf, Principles of Optics (7th edition, Cambridge University Press, Cambridge, 1999).

J. W. Goodman, and R. W. Lawrence, "Digital image formation from electronically detected holograms" Appl. Phys. Lett. 11, 77 (1967).

L. Onural, and M. Kocatepe, "Family of scaling chirp functions, diffraction, and holography" IEEE T. Signal Proces. 43, 1568-1578 (1995).

L. Cheng, "Simultaneous measurement of displacement and its spatial derivatives with a digital holographic method" Opt. Eng. 42, 3443-3446 (2003).

D. Gabor, "A new microscopic principle" Nature 61, 777-778 (1948).

J. W. Goodman, Introduction to Fourier Optics (3rd edition, Roberts and Company Publishers, Colorado, 2005).

U. Schnars and W. Jüptner, "Direct recording of holograms by a CCD target and numerical reconstruction" Appl. Opt. 33, 179-181 (1994).

F. Slimani, G. Grehan, G. Gouesbet, and D. Allano, "Near-field Lorenz-Mie theory and its application to microholography" Appl. Opt. 23, 4140-4148 (1984).

F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, "Suppression of the Moiré effect in sub-picosecond digital in-line holography" Opt. Express 15, 887-895 (2007).

M. Brunel, S. Coëtmellec, D. Lebrun, and K. A. Ameur, "Digital phase contrast with the fractional Fourier transform" Appl. Opt. 48, 579-583 (2009).

C. Buraga-Lefebvre, S. Coëtmellec, D. Lebrun, and C. Özkul, "Application of wavelet transform to hologram analysis: threedimensional location of particles" Opt. Lasers Eng. 33, 409-421 (2000).

S. Coëtmellec, D. Lebrun, and C. Özkul, "Application of the twodimensional fractional-order Fourier transformation to particle field digital holography" J. Opt. Soc. Am. A. 19, 1537-1546 (2002).

M. Malek, S. Coëtmellec, D. Lebrun, and D. Allano, "Formulation of in-line holography process by a linear shift invariant system: Application to the measurement of fiber diameter" Opt. Commun. 223, 263-271 (2003).

L. Onural, and P. D. Scott, "Digital decoding of in-line holograms" Opt. Eng. 26, 1124-1132 (1987).

F. Dubois, C. Schockaert, N. Callens, and C. Yourassowsky, "Focus plane detection criteria in digital holography microscopy by amplitude analysis" Opt. Express 14, 5895-5908 (2006).

W. Li, N. C. Loomis, Q. Hu, and C. S. Davis, "Focus detection from digital in-line holograms based on spectral l1 norms" J. Opt. Soc. Am. A. 24, 3054-3062 (2007).

A. J. E. M. Janssen, J. J. M. Braat, and P. Dirksen, "On the computation of the Nijboer-Zernike aberration integrals at arbitrary defocus" J. Mod. Opt. 51, 687-703 (2004).

M. Abramowitz, and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications Inc., New York, 1970).

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (1st edition, Wiley, England, 2001).