Abstract

The concept of numerical parametric lenses (NPL) is introduced to achieve wavefront reconstruction in digital holography. It is shown that operations usually performed by optical components and described in ray geometrical optics, such as image shifting, magnification, and especially complete aberration compensation (phase aberrations and image distortion), can be mimicked by numerical computation of a NPL. Furthermore, we demonstrate that automatic one-dimensional or two-dimensional fitting procedures allow adjustment of the NPL parameters as expressed in terms of standard or Zernike polynomial coefficients. These coefficients can provide a quantitative evaluation of the aberrations generated by the specimen. Demonstration is given of the reconstruction of the topology of a microlens.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. U. Schnars and W. P. O. Jüptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
    [CrossRef]
  2. E. Cuche, F. Bevilacqua, and C. Depeursinge, "Digital holography for quantitative phase-contrast imaging," Opt. Lett. 24, 291-293 (1999).
    [CrossRef]
  3. 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]
  4. P. Ferraro, S. De Nicola, A. Finizio, G. Coppola, S. Grilli, C. Magro, and G. Pierattini, "Compensation of the inherent wave front curvature in digital holographic coherent microscopy for quantitative phase-contrast imaging," Appl. Opt. 42, 1938-1946 (2003).
    [CrossRef] [PubMed]
  5. G. Pedrini, S. Schedin, and H. J. Tiziani, "Aberration compensation in digital holographic reconstruction of microscopic objects," J. Mod. Opt. 48, 1035-1041 (2001).
  6. A. Stadelmaier and J. H. Massig, "Compensation of lens aberrations in digital holography," Opt. Lett. 25, 1630-1632 (2000).
    [CrossRef]
  7. S. De Nicola, A. Finizio, G. Pierattini, D. Alfieri, S. Grilli, L. Sansone, and P. Ferraro, "Recovering correct phase information in multiwavelength digital holographic microscopy by compensation for chromatic aberrations," Opt. Lett. 30, 2706-2708 (2005).
    [CrossRef] [PubMed]
  8. 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," Opt. Lasers Eng. 37, 331-340 (2002).
    [CrossRef]
  9. S. Grilli, P. Ferraro, S. De Nicola, A. Finizio, G. Pierattini, and R. Meucci, "Whole optical wavefields reconstruction by digital holography," Opt. Express 9, 294-302 (2001).
    [CrossRef] [PubMed]
  10. 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. 26974-976 (2001).
    [CrossRef]
  11. S. De Nicola, A. Finizio, G. Pierattini, P. Ferraro, and D. Alfieri, "Angular spectrum method with correction of anamorphism for numerical reconstruction of digital holograms on tilted planes," Opt. Express 13, 9935-9940 (2005).
    [CrossRef] [PubMed]
  12. S. De Nicola, P. Ferraro, A. Finizio, S. Grilli, and G. Pierattini, "Experimental demonstration of the longitudinal image shift in digital holography," Opt. Eng. (Bellingham) 42, 1625-1630 (2003).
    [CrossRef]
  13. T. Colomb, E. Cuche, F. Charrière, J. Kühn, N. Aspert, F. Montfort, 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] [PubMed]
  14. P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, "Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms," Opt. Lett. 29, 854-856 (2004).
    [CrossRef] [PubMed]
  15. J. Kato, I. Yamaguchi, and T. Matsumura, "Multicolor digital holography with an achromatic phase shifter," Opt. Lett. 27, 1403-1405 (2002).
    [CrossRef]
  16. I. Yamaguchi, T. Matsumura, and J. Kato, "Phase-shifting color digital holography," Opt. Lett. 27, 1108-1110 (2002).
    [CrossRef]
  17. P. Almoro, M. Cadatal, W. Garcia, and C. Saloma, "Pulsed full-color digital holography with a hydrogen Raman shifter," Appl. Opt. 43, 2267-2271 (2004).
    [CrossRef] [PubMed]
  18. B. Javidi, P. Ferraro, S.-H Hong, S. D. Nicola, A. Finizio, D. Alfieri, and G. Pierattini, "Three-dimensional image fusion by use of multiwavelength digital holography," Opt. Lett. 30, 144-146 (2005).
    [CrossRef] [PubMed]
  19. G. Indebetouw and P. Klysubun, "Optical sectioning with low coherence spatiotemporal holography," Opt. Commun. 172, 25-29 (1999).
    [CrossRef]
  20. M. K. Kim, "Tomographic three-dimensional imaging of a biological specimen using wavelength-scanning digital interference holography," Opt. Express 7, 305-310 (2000).
    [CrossRef] [PubMed]
  21. A. Dakoff, J. Gass, and M. K. Kim, "Microscopic three-dimensional imaging by digital interference holography," J. Electron. Imaging 12, 643-647 (2003).
    [CrossRef]
  22. P. Massatsch, F. Charrière, E. Cuche, P. Marquet, and C. Depeursinge, "Time-domain optical coherence tomography with digital holographic microscopy," Appl. Opt. 44, 1806-1812 (2005).
    [CrossRef] [PubMed]
  23. A. Thelen, J. Bongartz, D. Giel, S. Frey, and P. Hering, "Iterative focus detection in hologram tomography," J. Opt. Soc. Am. A 22, 1176-1180 (2005).
    [CrossRef]
  24. L. Martínez-León, G. Pedrini, and W. Osten, "Applications of short-coherence digital holography in microscopy," Appl. Opt. 44, 3977-3984 (2005).
    [CrossRef] [PubMed]
  25. L. Yu and M. K. Kim, "Wavelength scanning digital interference holography for variable tomographic scanning," Opt. Express 13, 5621-5627 (2005).
    [CrossRef] [PubMed]
  26. F. Charrière, A. Marian, F. Montfort, J. Kühn, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, "Cell refractive index tomography by digital holographic microscopy," Opt. Lett. 31, 178-180 (2006).
    [CrossRef] [PubMed]
  27. F. C. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, "Algorithm for reconstruction of digital holograms with adjustable magnification," Opt. Lett. 29, 1668-1670 (2004).
    [CrossRef] [PubMed]
  28. F. Montfort, "Tomography using multiple wavelengths in digital holographic microscopy," Ph.D. dissertation (Ecole Polytechnique Fédérale de Lausanne, Lausanne, 2005).
  29. E. Cuche, P. Marquet, and C. Depeursinge, "Spatial filtering for zero-order and twin-image elimination in digital off-axis holography," Appl. Opt. 39, 4070-4075 (2000).
    [CrossRef]
  30. "ZEMAX: Optical Design Program, User's Guide, Version 10.0" (Focus Software, Tucson, 2001), pp. 126-127.
  31. 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] [PubMed]
  32. B. E. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
    [CrossRef]
  33. T. Colomb, F. Dürr, E. Cuche, P. Marquet, H. Limberger, R.-P. Salathé, and C. Depeursinge, "Polarization microscopy by use of digital holography: application to optical fiber birefringence measurements," Appl. Opt. 44, 4461-4469 (2005).
    [CrossRef] [PubMed]
  34. F. Montfort, T. Colomb, F. Charrière, J. Kühn, P. Marquet, E. Cuche, S. Herminjard, and C. Depeursinge, "Submicrometer optical tomography by multiple wavelength digital holographic microscopy," Appl. Opt. (to be published).

2006

2005

B. Javidi, P. Ferraro, S.-H Hong, S. D. Nicola, A. Finizio, D. Alfieri, and G. Pierattini, "Three-dimensional image fusion by use of multiwavelength digital holography," Opt. Lett. 30, 144-146 (2005).
[CrossRef] [PubMed]

P. Massatsch, F. Charrière, E. Cuche, P. Marquet, and C. Depeursinge, "Time-domain optical coherence tomography with digital holographic microscopy," Appl. Opt. 44, 1806-1812 (2005).
[CrossRef] [PubMed]

A. Thelen, J. Bongartz, D. Giel, S. Frey, and P. Hering, "Iterative focus detection in hologram tomography," J. Opt. Soc. Am. A 22, 1176-1180 (2005).
[CrossRef]

L. Martínez-León, G. Pedrini, and W. Osten, "Applications of short-coherence digital holography in microscopy," Appl. Opt. 44, 3977-3984 (2005).
[CrossRef] [PubMed]

T. Colomb, F. Dürr, E. Cuche, P. Marquet, H. Limberger, R.-P. Salathé, and C. Depeursinge, "Polarization microscopy by use of digital holography: application to optical fiber birefringence measurements," Appl. Opt. 44, 4461-4469 (2005).
[CrossRef] [PubMed]

L. Yu and M. K. Kim, "Wavelength scanning digital interference holography for variable tomographic scanning," Opt. Express 13, 5621-5627 (2005).
[CrossRef] [PubMed]

S. De Nicola, A. Finizio, G. Pierattini, D. Alfieri, S. Grilli, L. Sansone, and P. Ferraro, "Recovering correct phase information in multiwavelength digital holographic microscopy by compensation for chromatic aberrations," Opt. Lett. 30, 2706-2708 (2005).
[CrossRef] [PubMed]

S. De Nicola, A. Finizio, G. Pierattini, P. Ferraro, and D. Alfieri, "Angular spectrum method with correction of anamorphism for numerical reconstruction of digital holograms on tilted planes," Opt. Express 13, 9935-9940 (2005).
[CrossRef] [PubMed]

2004

2003

P. Ferraro, S. De Nicola, A. Finizio, G. Coppola, S. Grilli, C. Magro, and G. Pierattini, "Compensation of the inherent wave front curvature in digital holographic coherent microscopy for quantitative phase-contrast imaging," Appl. Opt. 42, 1938-1946 (2003).
[CrossRef] [PubMed]

S. De Nicola, P. Ferraro, A. Finizio, S. Grilli, and G. Pierattini, "Experimental demonstration of the longitudinal image shift in digital holography," Opt. Eng. (Bellingham) 42, 1625-1630 (2003).
[CrossRef]

A. Dakoff, J. Gass, and M. K. Kim, "Microscopic three-dimensional imaging by digital interference holography," J. Electron. Imaging 12, 643-647 (2003).
[CrossRef]

2002

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

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," Opt. Lasers Eng. 37, 331-340 (2002).
[CrossRef]

I. Yamaguchi, T. Matsumura, and J. Kato, "Phase-shifting color digital holography," Opt. Lett. 27, 1108-1110 (2002).
[CrossRef]

J. Kato, I. Yamaguchi, and T. Matsumura, "Multicolor digital holography with an achromatic phase shifter," Opt. Lett. 27, 1403-1405 (2002).
[CrossRef]

2001

2000

1999

Alfieri, D.

Almoro, P.

Aspert, N.

Bevilacqua, F.

Bongartz, J.

Cadatal, M.

Charrière, F.

Colomb, T.

Coppola, G.

Cuche, E.

T. Colomb, E. Cuche, F. Charrière, J. Kühn, N. Aspert, F. Montfort, 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] [PubMed]

F. Charrière, A. Marian, F. Montfort, J. Kühn, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, "Cell refractive index tomography by digital holographic microscopy," Opt. Lett. 31, 178-180 (2006).
[CrossRef] [PubMed]

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] [PubMed]

T. Colomb, F. Dürr, E. Cuche, P. Marquet, H. Limberger, R.-P. Salathé, and C. Depeursinge, "Polarization microscopy by use of digital holography: application to optical fiber birefringence measurements," Appl. Opt. 44, 4461-4469 (2005).
[CrossRef] [PubMed]

P. Massatsch, F. Charrière, E. Cuche, P. Marquet, and C. Depeursinge, "Time-domain optical coherence tomography with digital holographic microscopy," Appl. Opt. 44, 1806-1812 (2005).
[CrossRef] [PubMed]

E. Cuche, P. Marquet, and C. Depeursinge, "Spatial filtering for zero-order and twin-image elimination in digital off-axis holography," Appl. Opt. 39, 4070-4075 (2000).
[CrossRef]

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]

E. Cuche, F. Bevilacqua, and C. Depeursinge, "Digital holography for quantitative phase-contrast imaging," Opt. Lett. 24, 291-293 (1999).
[CrossRef]

F. Montfort, T. Colomb, F. Charrière, J. Kühn, P. Marquet, E. Cuche, S. Herminjard, and C. Depeursinge, "Submicrometer optical tomography by multiple wavelength digital holographic microscopy," Appl. Opt. (to be published).

Dakoff, A.

A. Dakoff, J. Gass, and M. K. Kim, "Microscopic three-dimensional imaging by digital interference holography," J. Electron. Imaging 12, 643-647 (2003).
[CrossRef]

Depeursinge, C.

T. Colomb, E. Cuche, F. Charrière, J. Kühn, N. Aspert, F. Montfort, 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] [PubMed]

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] [PubMed]

F. Charrière, A. Marian, F. Montfort, J. Kühn, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, "Cell refractive index tomography by digital holographic microscopy," Opt. Lett. 31, 178-180 (2006).
[CrossRef] [PubMed]

P. Massatsch, F. Charrière, E. Cuche, P. Marquet, and C. Depeursinge, "Time-domain optical coherence tomography with digital holographic microscopy," Appl. Opt. 44, 1806-1812 (2005).
[CrossRef] [PubMed]

T. Colomb, F. Dürr, E. Cuche, P. Marquet, H. Limberger, R.-P. Salathé, and C. Depeursinge, "Polarization microscopy by use of digital holography: application to optical fiber birefringence measurements," Appl. Opt. 44, 4461-4469 (2005).
[CrossRef] [PubMed]

E. Cuche, P. Marquet, and C. Depeursinge, "Spatial filtering for zero-order and twin-image elimination in digital off-axis holography," Appl. Opt. 39, 4070-4075 (2000).
[CrossRef]

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]

E. Cuche, F. Bevilacqua, and C. Depeursinge, "Digital holography for quantitative phase-contrast imaging," Opt. Lett. 24, 291-293 (1999).
[CrossRef]

F. Montfort, T. Colomb, F. Charrière, J. Kühn, P. Marquet, E. Cuche, S. Herminjard, and C. Depeursinge, "Submicrometer optical tomography by multiple wavelength digital holographic microscopy," Appl. Opt. (to be published).

Dürr, F.

Emery, Y.

Ferraro, P.

B. Javidi, P. Ferraro, S.-H Hong, S. D. Nicola, A. Finizio, D. Alfieri, and G. Pierattini, "Three-dimensional image fusion by use of multiwavelength digital holography," Opt. Lett. 30, 144-146 (2005).
[CrossRef] [PubMed]

S. De Nicola, A. Finizio, G. Pierattini, P. Ferraro, and D. Alfieri, "Angular spectrum method with correction of anamorphism for numerical reconstruction of digital holograms on tilted planes," Opt. Express 13, 9935-9940 (2005).
[CrossRef] [PubMed]

S. De Nicola, A. Finizio, G. Pierattini, D. Alfieri, S. Grilli, L. Sansone, and P. Ferraro, "Recovering correct phase information in multiwavelength digital holographic microscopy by compensation for chromatic aberrations," Opt. Lett. 30, 2706-2708 (2005).
[CrossRef] [PubMed]

P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, "Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms," Opt. Lett. 29, 854-856 (2004).
[CrossRef] [PubMed]

S. De Nicola, P. Ferraro, A. Finizio, S. Grilli, and G. Pierattini, "Experimental demonstration of the longitudinal image shift in digital holography," Opt. Eng. (Bellingham) 42, 1625-1630 (2003).
[CrossRef]

P. Ferraro, S. De Nicola, A. Finizio, G. Coppola, S. Grilli, C. Magro, and G. Pierattini, "Compensation of the inherent wave front curvature in digital holographic coherent microscopy for quantitative phase-contrast imaging," Appl. Opt. 42, 1938-1946 (2003).
[CrossRef] [PubMed]

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," Opt. Lasers Eng. 37, 331-340 (2002).
[CrossRef]

S. Grilli, P. Ferraro, S. De Nicola, A. Finizio, G. Pierattini, and R. Meucci, "Whole optical wavefields reconstruction by digital holography," Opt. Express 9, 294-302 (2001).
[CrossRef] [PubMed]

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. 26974-976 (2001).
[CrossRef]

Finizio, A.

S. De Nicola, A. Finizio, G. Pierattini, P. Ferraro, and D. Alfieri, "Angular spectrum method with correction of anamorphism for numerical reconstruction of digital holograms on tilted planes," Opt. Express 13, 9935-9940 (2005).
[CrossRef] [PubMed]

S. De Nicola, A. Finizio, G. Pierattini, D. Alfieri, S. Grilli, L. Sansone, and P. Ferraro, "Recovering correct phase information in multiwavelength digital holographic microscopy by compensation for chromatic aberrations," Opt. Lett. 30, 2706-2708 (2005).
[CrossRef] [PubMed]

B. Javidi, P. Ferraro, S.-H Hong, S. D. Nicola, A. Finizio, D. Alfieri, and G. Pierattini, "Three-dimensional image fusion by use of multiwavelength digital holography," Opt. Lett. 30, 144-146 (2005).
[CrossRef] [PubMed]

P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, "Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms," Opt. Lett. 29, 854-856 (2004).
[CrossRef] [PubMed]

S. De Nicola, P. Ferraro, A. Finizio, S. Grilli, and G. Pierattini, "Experimental demonstration of the longitudinal image shift in digital holography," Opt. Eng. (Bellingham) 42, 1625-1630 (2003).
[CrossRef]

P. Ferraro, S. De Nicola, A. Finizio, G. Coppola, S. Grilli, C. Magro, and G. Pierattini, "Compensation of the inherent wave front curvature in digital holographic coherent microscopy for quantitative phase-contrast imaging," Appl. Opt. 42, 1938-1946 (2003).
[CrossRef] [PubMed]

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," Opt. Lasers Eng. 37, 331-340 (2002).
[CrossRef]

S. Grilli, P. Ferraro, S. De Nicola, A. Finizio, G. Pierattini, and R. Meucci, "Whole optical wavefields reconstruction by digital holography," Opt. Express 9, 294-302 (2001).
[CrossRef] [PubMed]

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. 26974-976 (2001).
[CrossRef]

Frey, S.

Garcia, W.

Gass, J.

A. Dakoff, J. Gass, and M. K. Kim, "Microscopic three-dimensional imaging by digital interference holography," J. Electron. Imaging 12, 643-647 (2003).
[CrossRef]

Giel, D.

Grilli, S.

Hering, P.

Herminjard, S.

F. Montfort, T. Colomb, F. Charrière, J. Kühn, P. Marquet, E. Cuche, S. Herminjard, and C. Depeursinge, "Submicrometer optical tomography by multiple wavelength digital holographic microscopy," Appl. Opt. (to be published).

Hong, S.-H

Indebetouw, G.

G. Indebetouw and P. Klysubun, "Optical sectioning with low coherence spatiotemporal holography," Opt. Commun. 172, 25-29 (1999).
[CrossRef]

Javidi, B.

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]

Kato, J.

Kim, M. K.

Klysubun, P.

G. Indebetouw and P. Klysubun, "Optical sectioning with low coherence spatiotemporal holography," Opt. Commun. 172, 25-29 (1999).
[CrossRef]

Kühn, J.

Limberger, H.

Magro, C.

Marian, A.

Marquet, P.

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] [PubMed]

F. Charrière, A. Marian, F. Montfort, J. Kühn, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, "Cell refractive index tomography by digital holographic microscopy," Opt. Lett. 31, 178-180 (2006).
[CrossRef] [PubMed]

T. Colomb, E. Cuche, F. Charrière, J. Kühn, N. Aspert, F. Montfort, 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] [PubMed]

P. Massatsch, F. Charrière, E. Cuche, P. Marquet, and C. Depeursinge, "Time-domain optical coherence tomography with digital holographic microscopy," Appl. Opt. 44, 1806-1812 (2005).
[CrossRef] [PubMed]

T. Colomb, F. Dürr, E. Cuche, P. Marquet, H. Limberger, R.-P. Salathé, and C. Depeursinge, "Polarization microscopy by use of digital holography: application to optical fiber birefringence measurements," Appl. Opt. 44, 4461-4469 (2005).
[CrossRef] [PubMed]

E. Cuche, P. Marquet, and C. Depeursinge, "Spatial filtering for zero-order and twin-image elimination in digital off-axis holography," Appl. Opt. 39, 4070-4075 (2000).
[CrossRef]

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]

F. Montfort, T. Colomb, F. Charrière, J. Kühn, P. Marquet, E. Cuche, S. Herminjard, and C. Depeursinge, "Submicrometer optical tomography by multiple wavelength digital holographic microscopy," Appl. Opt. (to be published).

Martínez-León, L.

Massatsch, P.

Massig, J. H.

Matsumura, T.

Meucci, R.

Monfort, F.

Montfort, F.

F. Charrière, A. Marian, F. Montfort, J. Kühn, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, "Cell refractive index tomography by digital holographic microscopy," Opt. Lett. 31, 178-180 (2006).
[CrossRef] [PubMed]

T. Colomb, E. Cuche, F. Charrière, J. Kühn, N. Aspert, F. Montfort, 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] [PubMed]

F. Montfort, T. Colomb, F. Charrière, J. Kühn, P. Marquet, E. Cuche, S. Herminjard, and C. Depeursinge, "Submicrometer optical tomography by multiple wavelength digital holographic microscopy," Appl. Opt. (to be published).

F. Montfort, "Tomography using multiple wavelengths in digital holographic microscopy," Ph.D. dissertation (Ecole Polytechnique Fédérale de Lausanne, Lausanne, 2005).

Nicola, S. D.

Nicola, S. De

S. De Nicola, A. Finizio, G. Pierattini, P. Ferraro, and D. Alfieri, "Angular spectrum method with correction of anamorphism for numerical reconstruction of digital holograms on tilted planes," Opt. Express 13, 9935-9940 (2005).
[CrossRef] [PubMed]

S. De Nicola, A. Finizio, G. Pierattini, D. Alfieri, S. Grilli, L. Sansone, and P. Ferraro, "Recovering correct phase information in multiwavelength digital holographic microscopy by compensation for chromatic aberrations," Opt. Lett. 30, 2706-2708 (2005).
[CrossRef] [PubMed]

P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, "Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms," Opt. Lett. 29, 854-856 (2004).
[CrossRef] [PubMed]

S. De Nicola, P. Ferraro, A. Finizio, S. Grilli, and G. Pierattini, "Experimental demonstration of the longitudinal image shift in digital holography," Opt. Eng. (Bellingham) 42, 1625-1630 (2003).
[CrossRef]

P. Ferraro, S. De Nicola, A. Finizio, G. Coppola, S. Grilli, C. Magro, and G. Pierattini, "Compensation of the inherent wave front curvature in digital holographic coherent microscopy for quantitative phase-contrast imaging," Appl. Opt. 42, 1938-1946 (2003).
[CrossRef] [PubMed]

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," Opt. Lasers Eng. 37, 331-340 (2002).
[CrossRef]

S. Grilli, P. Ferraro, S. De Nicola, A. Finizio, G. Pierattini, and R. Meucci, "Whole optical wavefields reconstruction by digital holography," Opt. Express 9, 294-302 (2001).
[CrossRef] [PubMed]

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. 26974-976 (2001).
[CrossRef]

Osten, W.

Pedrini, G.

L. Martínez-León, G. Pedrini, and W. Osten, "Applications of short-coherence digital holography in microscopy," Appl. Opt. 44, 3977-3984 (2005).
[CrossRef] [PubMed]

G. Pedrini, S. Schedin, and H. J. Tiziani, "Aberration compensation in digital holographic reconstruction of microscopic objects," J. Mod. Opt. 48, 1035-1041 (2001).

Pierattini, G.

B. Javidi, P. Ferraro, S.-H Hong, S. D. Nicola, A. Finizio, D. Alfieri, and G. Pierattini, "Three-dimensional image fusion by use of multiwavelength digital holography," Opt. Lett. 30, 144-146 (2005).
[CrossRef] [PubMed]

S. De Nicola, A. Finizio, G. Pierattini, P. Ferraro, and D. Alfieri, "Angular spectrum method with correction of anamorphism for numerical reconstruction of digital holograms on tilted planes," Opt. Express 13, 9935-9940 (2005).
[CrossRef] [PubMed]

S. De Nicola, A. Finizio, G. Pierattini, D. Alfieri, S. Grilli, L. Sansone, and P. Ferraro, "Recovering correct phase information in multiwavelength digital holographic microscopy by compensation for chromatic aberrations," Opt. Lett. 30, 2706-2708 (2005).
[CrossRef] [PubMed]

P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, "Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms," Opt. Lett. 29, 854-856 (2004).
[CrossRef] [PubMed]

S. De Nicola, P. Ferraro, A. Finizio, S. Grilli, and G. Pierattini, "Experimental demonstration of the longitudinal image shift in digital holography," Opt. Eng. (Bellingham) 42, 1625-1630 (2003).
[CrossRef]

P. Ferraro, S. De Nicola, A. Finizio, G. Coppola, S. Grilli, C. Magro, and G. Pierattini, "Compensation of the inherent wave front curvature in digital holographic coherent microscopy for quantitative phase-contrast imaging," Appl. Opt. 42, 1938-1946 (2003).
[CrossRef] [PubMed]

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," Opt. Lasers Eng. 37, 331-340 (2002).
[CrossRef]

S. Grilli, P. Ferraro, S. De Nicola, A. Finizio, G. Pierattini, and R. Meucci, "Whole optical wavefields reconstruction by digital holography," Opt. Express 9, 294-302 (2001).
[CrossRef] [PubMed]

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. 26974-976 (2001).
[CrossRef]

Salathé, R.-P.

Saleh, B. E.

B. E. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

Saloma, C.

Sansone, L.

Schedin, S.

G. Pedrini, S. Schedin, and H. J. Tiziani, "Aberration compensation in digital holographic reconstruction of microscopic objects," J. Mod. Opt. 48, 1035-1041 (2001).

Schnars, U.

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

Stadelmaier, A.

Teich, M. C.

B. E. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

Thelen, A.

Tiziani, H. J.

G. Pedrini, S. Schedin, and H. J. Tiziani, "Aberration compensation in digital holographic reconstruction of microscopic objects," J. Mod. Opt. 48, 1035-1041 (2001).

Weible, K.

Yamaguchi, I.

Yaroslavsky, L. P.

Yu, L.

Zhang, F. C.

Appl. Opt.

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]

E. Cuche, P. Marquet, and C. Depeursinge, "Spatial filtering for zero-order and twin-image elimination in digital off-axis holography," Appl. Opt. 39, 4070-4075 (2000).
[CrossRef]

P. Ferraro, S. De Nicola, A. Finizio, G. Coppola, S. Grilli, C. Magro, and G. Pierattini, "Compensation of the inherent wave front curvature in digital holographic coherent microscopy for quantitative phase-contrast imaging," Appl. Opt. 42, 1938-1946 (2003).
[CrossRef] [PubMed]

P. Almoro, M. Cadatal, W. Garcia, and C. Saloma, "Pulsed full-color digital holography with a hydrogen Raman shifter," Appl. Opt. 43, 2267-2271 (2004).
[CrossRef] [PubMed]

P. Massatsch, F. Charrière, E. Cuche, P. Marquet, and C. Depeursinge, "Time-domain optical coherence tomography with digital holographic microscopy," Appl. Opt. 44, 1806-1812 (2005).
[CrossRef] [PubMed]

L. Martínez-León, G. Pedrini, and W. Osten, "Applications of short-coherence digital holography in microscopy," Appl. Opt. 44, 3977-3984 (2005).
[CrossRef] [PubMed]

T. Colomb, F. Dürr, E. Cuche, P. Marquet, H. Limberger, R.-P. Salathé, and C. Depeursinge, "Polarization microscopy by use of digital holography: application to optical fiber birefringence measurements," Appl. Opt. 44, 4461-4469 (2005).
[CrossRef] [PubMed]

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] [PubMed]

T. Colomb, E. Cuche, F. Charrière, J. Kühn, N. Aspert, F. Montfort, 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] [PubMed]

J. Electron. Imaging

A. Dakoff, J. Gass, and M. K. Kim, "Microscopic three-dimensional imaging by digital interference holography," J. Electron. Imaging 12, 643-647 (2003).
[CrossRef]

J. Mod. Opt.

G. Pedrini, S. Schedin, and H. J. Tiziani, "Aberration compensation in digital holographic reconstruction of microscopic objects," J. Mod. Opt. 48, 1035-1041 (2001).

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]

Opt. Commun.

G. Indebetouw and P. Klysubun, "Optical sectioning with low coherence spatiotemporal holography," Opt. Commun. 172, 25-29 (1999).
[CrossRef]

Opt. Eng. (Bellingham)

S. De Nicola, P. Ferraro, A. Finizio, S. Grilli, and G. Pierattini, "Experimental demonstration of the longitudinal image shift in digital holography," Opt. Eng. (Bellingham) 42, 1625-1630 (2003).
[CrossRef]

Opt. Express

Opt. Lasers Eng.

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," Opt. Lasers Eng. 37, 331-340 (2002).
[CrossRef]

Opt. Lett.

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. 26974-976 (2001).
[CrossRef]

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

P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, "Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms," Opt. Lett. 29, 854-856 (2004).
[CrossRef] [PubMed]

F. C. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, "Algorithm for reconstruction of digital holograms with adjustable magnification," Opt. Lett. 29, 1668-1670 (2004).
[CrossRef] [PubMed]

B. Javidi, P. Ferraro, S.-H Hong, S. D. Nicola, A. Finizio, D. Alfieri, and G. Pierattini, "Three-dimensional image fusion by use of multiwavelength digital holography," Opt. Lett. 30, 144-146 (2005).
[CrossRef] [PubMed]

F. Charrière, A. Marian, F. Montfort, J. Kühn, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, "Cell refractive index tomography by digital holographic microscopy," Opt. Lett. 31, 178-180 (2006).
[CrossRef] [PubMed]

E. Cuche, F. Bevilacqua, and C. Depeursinge, "Digital holography for quantitative phase-contrast imaging," Opt. Lett. 24, 291-293 (1999).
[CrossRef]

S. De Nicola, A. Finizio, G. Pierattini, D. Alfieri, S. Grilli, L. Sansone, and P. Ferraro, "Recovering correct phase information in multiwavelength digital holographic microscopy by compensation for chromatic aberrations," Opt. Lett. 30, 2706-2708 (2005).
[CrossRef] [PubMed]

I. Yamaguchi, T. Matsumura, and J. Kato, "Phase-shifting color digital holography," Opt. Lett. 27, 1108-1110 (2002).
[CrossRef]

J. Kato, I. Yamaguchi, and T. Matsumura, "Multicolor digital holography with an achromatic phase shifter," Opt. Lett. 27, 1403-1405 (2002).
[CrossRef]

Other

F. Montfort, "Tomography using multiple wavelengths in digital holographic microscopy," Ph.D. dissertation (Ecole Polytechnique Fédérale de Lausanne, Lausanne, 2005).

"ZEMAX: Optical Design Program, User's Guide, Version 10.0" (Focus Software, Tucson, 2001), pp. 126-127.

B. E. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

F. Montfort, T. Colomb, F. Charrière, J. Kühn, P. Marquet, E. Cuche, S. Herminjard, and C. Depeursinge, "Submicrometer optical tomography by multiple wavelength digital holographic microscopy," Appl. Opt. (to be published).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (22)

Fig. 1
Fig. 1

Digital holographic microscope, (a) transmission and (b) reflection setups. O object wave; R reference wave; BS beam splitter; M1, M2 mirrors; MO microscope objective, RL lens in the reference wave, OC condenser in the object wave. For demonstration purposes a tilted plate is introduced between the BS and the CCD to intentionally produce aberrations. (c) Detail of the off-axis geometry.

Fig. 2
Fig. 2

Two-dimensional fitting procedure with standard polynomial model (left column) and Zernike polynomial model (right), (a) the reconstructed amplitude contrast with the assumed flat areas F situated inside the white rectangles. (b) and (f) reconstruction with initial parameters computed with 1D procedure. (c) and (g) 2D unwrap of (b) and (f), respectively (d) and (e) respectively, the corrections with six and ten adjusted standard polynomial coefficients ( o = 2 , 3 ) . (h) and (i) respectively, the correction with six and eight adjusted Zernike polynomial coefficients ( o = 5 , 7 ) .

Fig. 3
Fig. 3

(a) Microlens phase by applying 2D fitting procedure with Zernike polynomial on points included in areas indicated by white lines. (b) Perspective representation of 2D phase unwrap of (a). (c)–(e) Microlens shape compensation with Zernike formulation with (c) 10 parameters, (d) 11 parameters, (e) 21 parameters.

Fig. 4
Fig. 4

Repartition of Zernike coefficients for an adjustment of 21 coefficients. The absolute coefficient values are plotted. Black and gray patterns indicate, respectively, negative and positive values.

Fig. 5
Fig. 5

Procedure of spectrum centering. (a) Initial filtered spectrum, (b) spectrum centered. The arrow represents the shift between the amplitude maximum of the frequencies associated with the virtual image and the center of the entire spectrum. (c) Spectrum of a hologram for which the curvatures of the reference and object waves are different, inducing a nonpunctual central frequency in the spectrum.

Fig. 6
Fig. 6

Principle of digital reconstruction process to center the ROI. (a) Hologram recording, (b) reconstruction with a digital reference wave U = 1 (the ROI is not centered), (c) reconstruction with a digital reference wave U = R (the ROI is centered).

Fig. 7
Fig. 7

Comparison between SFTF [(a)–(d)] and CF [(e)–(h)], with [(c), (d), (g), (h)] or without [(a), (b), (e), (f)] ROI centering. (a), (c), (e), (g) are amplitude images and (b), (d), (f), (h) the corresponding phase reconstructions.

Fig. 8
Fig. 8

Aliasing appears when the reconstruction distance is too small. (a) d = 11 cm , (b) d = 5 cm , (c) aliasing at d = 3.3 cm . With ROI centering, the reconstruction (d) can be achieved without aliasing.

Fig. 9
Fig. 9

Adjustment of the tilt parameters of the NPL Γ H by applying 1D procedure along black profiles. (a) The initial phase in the hologram plane, (b) the corresponding amplitude reconstruction in SFTF, (c) tilt-corrected phase in the hologram plane, (d) the corresponding centered amplitude reconstruction.

Fig. 10
Fig. 10

Shifting procedure: (a) and (b) show, respectively, the amplitude and phase reconstructions after tilt compensation. The arrows define the chosen translation of the ROI. (c) and (d) show the respective amplitude and phase shifted reconstructions.

Fig. 11
Fig. 11

H, hologram plane; I, image plane; Δ S x , chosen shift in the direction x; d, reconstruction distance; θ x , shifting angle.

Fig. 12
Fig. 12

Amplitude and phase reconstructions are presented, respectively, on the left and on the right. The reconstructions are done from a hologram recorded with (a), (b) λ 1 = 480 nm ; (c)–(f) λ 2 = 700 nm . The white rectangle defines the reference size. The white dashed rectangle defines the size of the same object without performing magnification. Images in (e), (f) are reconstructed from the same hologram as in (b), (c) after performing a magnification procedure M = 1.0038 defined by the ratio of the rectangle sizes.

Fig. 13
Fig. 13

(a), (b) Mean amplitude reconstructed from 20 holograms recorded with different wavelengths and (c), (d) mapping of it on the 3D topography of the specimen. Images in (b), (d) are processed with shift and magnification compensation.

Fig. 14
Fig. 14

(a) Correction of the tilt in the hologram plane and respective (b) amplitude and (c) phase reconstructions in SFTF at a distance d = 17.46 cm without NPL in image plane. (d) High-order correction in the hologram plane and respective (e) amplitude and (f) phase reconstruction at a distance d = 8.78 cm . The correction in the hologram plane is preserved along the direction of propagation and a numerical lens is no longer necessary in the image plane.

Fig. 15
Fig. 15

Hologram of USAF test target recorded with a cylindrical lens as MO.

Fig. 16
Fig. 16

Amplitude reconstruction for different distances of reconstruction. Because of the astigmatism of the cylindrical lens, there are two different focal points located at d = 5.5 cm and d = . The reconstructed image is focused at d = 23.3 cm .

Fig. 17
Fig. 17

Amplitude reconstruction with different parameters, (a) CF, M = 1 , d = 23.3 cm ; (b) SFTF, d = 23.3 cm , (c) CF, M = 0.3 , d = 6.99 . The astigmatism shown in detail in (c) is compensated by (d) the adjustment of P 20 H , A = 0.11 × 10 10 or by (e) defining two reconstruction distances d 1 = 6.99 cm and d 2 = 7.92 cm .

Fig. 18
Fig. 18

(a) Phase reconstruction with P 20 H , A = 0.11 × 10 10 without Γ I , C ; the other images are compensated with Γ I , C , (b) P 20 H , A = 0 , (c) P 20 H , A = 0.11 × 10 10 , and (d) two reconstruction distances. The black lines have the same length and reveal a dilatation in the image in the horizontal direction for (d).

Fig. 19
Fig. 19

Phase image in hologram plane: (a) without Γ S H , C adjustment, (b) after adjustment of standard polynomial order o = 3 . The straight black lines define the profiles used to set the initial values of 2D fitting parameters, and the curved white lines delimit the areas excluded from the areas known to be flat.

Fig. 20
Fig. 20

Amplitude (left) and phase (right) reconstructions after Γ S H , C adjustment: (a) d = 13.3 cm , (b) d = 6.9 cm , (c) d 1 = 13.3 cm and d 2 = 6.9 cm , (d) M = 0.56 ( d = 7.1 cm ) and P 02 H = 4.7 × 10 10 , (e) M = 0.56 ( d = 7.95 cm ) and P 20 H = 4.7 × 10 10 .

Fig. 21
Fig. 21

(a)–(c) The correction of the tilt is done in the hologram plane, and the aberration compensation is performed in the image plane. (d),(e) Compensation with Γ S H , C with seventh-order standard polynomial 2D fitting. (a), (d) Hologram, plane phase images. (b), (e) and (c), (f), respectively, amplitude and phase images in the image plane. The image distortion clearly visible in (b), (c) is compensated in (e), (f).

Fig. 22
Fig. 22

(a)–(c) Compensation with Γ S H , C with eighth-order standard polynomial 2D fitting; (d)–(f) after adding manual adjustment of primary spherical Zernike term Z 10 = 9.83 × 10 7 and a compensation of the resulting phase deformation in the image plane by an automatic adjustment of Γ S I , C (six orders). (a) and (d) show the hologram plane phase image; (b), (e) and (c), (f), respectively, show the amplitude and phase images in the image plane. The image distortion clearly visible in (b), (c) is totally compensated in (e), (f).

Tables (1)

Tables Icon

Table 1 Zernike Standard Coefficients in ZEMAX Classification

Equations (39)

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

I H ( x , y ) = ( R + O ) ( R + O ) * = R 2 + O 2 + R * O + RO * .
I H ( k , l ) = k Δ x Δ x 2 k Δ y + Δ y 2 l Δ y Δ y 2 l Δ y + Δ y 2 I H ( x , y ) d x d y ,
I H F = R * O .
Ψ SFTF ( m , n ) = Γ I ( m , n ) A exp [ i π λ d ( m 2 Δ ξ 2 + n 2 Δ η 2 ) ] × FFT { Γ H ( k , l ) I H F ( k , l ) exp [ i π λ d ( k 2 Δ x 2 + l 2 Δ y 2 ) ] } ,
Ψ CF ( m , n ) = Γ I ( m , n ) A × FFT 1 { FFT [ Γ H ( k , l ) I H F ( k , l ) ] exp [ i π λ d ( ν k 2 + ν l 2 ) ] } ,
Δ ξ = Δ η = λ d N Δ x .
α ξ = Δ x Δ ξ = N Δ x 2 λ d , α η = Δ y Δ η = N Δ y 2 λ d .
Γ S ( m , n ) = exp ( i 2 π λ α = β = 0 α + β = o P α β m α m β ) ,
Γ Z ( m , n ) = exp ( i 2 π λ α = 0 o P α Z α ) ,
Γ P = Γ S P , Sh Γ S P , M Γ PM P , C .
Y ( γ m , ζ n ) = α = β = 0 α + β = o a α β S α β , S α β = γ m α ζ n β ,
Y ( γ m , ζ n ) = α = 0 α = o a α Z α ,
M × A M = Y ,
P M ( i ) = P M ( i 1 ) + A M ( i ) .
Γ M P , C = exp [ i 2 π λ P M M ] .
P S ( 0 ) = [ 0 P 10 1 D P 01 1 D P 20 1 D P 02 1 D ] ,
P Z ( 0 ) = [ 0 P 10 1 D 2 P 01 1 D 2 ] ,
Δ S j = N S j Δ j ,
Γ S H , Sh ( x ) = exp [ i 2 π λ S ̂ x ] = exp [ i 2 π λ ( S x m Δ x + S y n Δ y ) ] ,
S j = sin ( θ j ) = sin [ arctan ( Δ S j d ) ] .
P 10 H , Sh = sin [ arctan ( Δ S x d ) ] Δ x ,
P 01 H , Sh = sin [ arctan ( Δ S y d ) ] Δ y .
Γ S I , Sh ( m , n ) = exp [ i 2 π λ ( P 10 I , Sh m + P 01 I , Sh n ) ] ,
θ θ max = arcsin ( λ 2 Δ x ) .
arctan ( Δ S j d ) θ max = arcsin ( λ 2 Δ x ) .
M = d ( d ) = d d .
1 f = 1 ( d ) + 1 d .
Γ H , M ( m , n ) = exp [ i 2 π λ 1 2 f ( m 2 Δ x 2 + n 2 Δ y 2 ) ] ,
Γ H , M ( m , n ) = exp [ i 2 π λ ( P 20 H , M m 2 + P 02 H , M n 2 ) ] ,
P 20 H , M = P 02 H , M = Δ x 2 2 f .
d = M d ,
P 02 H , M = P 20 H , M = ( 1 M 1 ) Δ x 2 2 d .
Γ I , M ( m , n ) = exp [ i 2 π λ 1 2 ( f d ) ( m 2 Δ x 2 + n 2 Δ y 2 ) ] ,
Γ I , M ( m , n ) = exp [ i 2 π λ ( P 20 I , M m 2 + P 02 I , M n 2 ) ] ,
P 20 I , M = P 02 I , M = Δ x 2 ( M 1 ) 2 M 2 d .
R ( x , y ) = R exp [ i ( k x x + k y y ) ] exp [ i W R ( x , y ) ] ,
O ( x , y ) = O exp [ i φ ( x , y ) ] exp [ i W O ( x , y ) ] .
I H F = R O exp [ i ( k x x + k y y ) ] exp [ i ( φ + W O W R ) ] .
P 20 I , A = Δ x 2 ( Δ x 2 2 P 20 H ) d .

Metrics