OSA's Digital Library

Applied Optics

Applied Optics


  • Vol. 42, Iss. 29 — Oct. 10, 2003
  • pp: 5816–5824

Imaging phase objects with square-root, Foucault, and Hoffman real filters: a comparison

Arkadiusz Sagan, Slawomir Nowicki, Ryszard Buczynski, Marek Kowalczyk, and Tomasz Szoplik  »View Author Affiliations

Applied Optics, Vol. 42, Issue 29, pp. 5816-5824 (2003)

View Full Text Article

Enhanced HTML    Acrobat PDF (823 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



Methods of imaging phase objects are considered. First the square-root filter is inferred from a definition of fractional-order derivatives given in terms of the integration of a fractional order called the Riemann-Liouville integral. Then we present a comparison of the performance of three frequency-domain real filters: square root, Foucault, and Hoffman. The phase-object imaging method is useful as a phase-shift measurement technique under the condition that the output image intensity is a known function of object phase. For the square-root filter it is the first derivative of the object phase function. The Foucault filter, in spite of its position, gives output image intensities expressed by Hilbert transforms. The output image intensity obtained with the Hoffman filter is not expressed by an analytical formula. The performance of the filters in a 4f imaging system with coherent illumination is simulated by use of VirtualLab 1.0 software.

© 2003 Optical Society of America

OCIS Codes
(070.0070) Fourier optics and signal processing : Fourier optics and signal processing
(070.1170) Fourier optics and signal processing : Analog optical signal processing
(070.6110) Fourier optics and signal processing : Spatial filtering
(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology
(120.5050) Instrumentation, measurement, and metrology : Phase measurement

Original Manuscript: December 13, 2002
Revised Manuscript: July 16, 2003
Published: October 10, 2003

Arkadiusz Sagan, Slawomir Nowicki, Ryszard Buczynski, Marek Kowalczyk, and Tomasz Szoplik, "Imaging phase objects with square-root, Foucault, and Hoffman real filters: a comparison," Appl. Opt. 42, 5816-5824 (2003)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. M. Kowalczyk, “Spectral and imaging properties of uniform diffusers,” J. Opt. Soc. Am. 1, 192–200 (1984). [CrossRef]
  2. G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook (SPIE Press, Bellingham, Wash., 1989), pp. 474–502. [CrossRef]
  3. J. Ojeda-Castaneda, “A proposal to classify methods employed to detect thin phase structures under coherent illumination,” Opt. Acta 27, 917–929 (1980). [CrossRef]
  4. R. A. Sprague, B. J. Thompson, “Quantitative visualization of large variation phase objects,” Appl. Opt. 11, 1469–1479 (1972). [CrossRef] [PubMed]
  5. H. Furuhashi, K. Matsuda, C. P. Grover, “Visualization of phase objects by use of a differentiation filter,” Appl. Opt. 42, 218–226 (2003). [CrossRef] [PubMed]
  6. A. W. Lohmann, J. Schwider, N. Streibl, J. Thomas, “Array illuminator based on phase contrast,” Appl. Opt. 27, 2915–2921 (1988). [CrossRef] [PubMed]
  7. C. S. Anderson, “Fringe visibility, irradiance, and accuracy in common path interferometers for visualization of phase disturbances,” Appl. Opt. 34, 7474–7485 (1995). [CrossRef] [PubMed]
  8. J. Glückstad, P. C. Mogensen, “Optimal phase contrast in common path interferometry,” Appl. Opt. 40, 268–282 (2001). [CrossRef]
  9. D. Sánchez-de-la-Llave, M. D. Iturbe Castillo, “Influence of illuminating beyond the object support on Zernike type phase contrast filtering,” Appl. Opt. 41, 2607–2612 (2002). [CrossRef] [PubMed]
  10. R. Hoffman, L. Gross, “Modulation contrast microscope,” Appl. Opt. 14, 1169–1176 (1975). [CrossRef] [PubMed]
  11. B. A. Horwitz, “Phase image differentiation with linear intensity output,” Appl. Opt. 17, 181–186 (1978). [CrossRef] [PubMed]
  12. J. Lancis, T. Szoplik, E. Tajahuerce, V. Climent, M. Fernández-Alonso, “Fractional derivative Fourier plane filter for phase-change visualization,” Appl. Opt. 36, 7461–7464 (1997). [CrossRef]
  13. E. Tajahuerce, T. Szoplik, J. Lancis, V. Climent, M. Fernández-Alonso, “Phase-object fractional differentiation using Fourier plane filters,” Pure Appl. Opt. 6, 481–490 (1997). [CrossRef]
  14. T. Szoplik, V. Climent, E. Tajahuerce, J. Lancis, M. Fernández-Alonso, “Phase-change visualization in two-dimensional phase objects with a semiderivative real filter,” Appl. Opt. 37, 5472–5478 (1998). [CrossRef]
  15. L. M. Soroko, Hilbert Optics (Science, Moscow, 1981), pp. 34–94 (in Russian).
  16. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1986).
  17. H. Kasprzak, “On the possibility of optical performing of non-integer order derivatives,” Opt. Appl. 10, 289–292 (1980).
  18. H. Kasprzak, “Differentiation of a noninteger order and its optical implementation,” Appl. Opt. 21, 3287–3291 (1982). [CrossRef] [PubMed]
  19. J. L. Lavoie, T. J. Osler, R. Tremblay, “Fractional derivatives and special functions,” SIAM Rev. 18, 240–267 (1976). [CrossRef]
  20. K. B. Oldham, J. Spanier, The Fractional Calculus (Academic, Orlando, Fla., 1974).
  21. S. G. Samko, A. A. Kilbas, O. I. Maritchev, Fractional Integrals and Derivatives and Their Applications (Science and Technique, Minsk, Russia, 1987; in Russian).
  22. G. S. Settles, Schlieren and Shadowgraph Techniques (Springer-Verlag, Berlin, 2001). [CrossRef]
  23. M. Pluta, Advanced Light Microscopy. Volume 2: Specialized Methods (PWN and Elsevier, Warsaw, 1989).
  24. E. W. S. Hee, “Fabrication of apodized apertures for laser beam attenuation,” Opt. Laser Technol. 7, 75–79 (1975). [CrossRef]
  25. J. A. Davis, D. A. Smith, D. E. McNamara, D. M. Cottrell, J. Campos, “Fractional derivatives—analysis and experimental implementation,” Appl. Opt. 40, 5943–5948 (2001). [CrossRef]
  26. J. A. Davis, M. D. Nowak, “Selective edge enhancement of images with an acousto-optic light modulator,” Appl. Opt. 41, 4835–4839 (2002). [CrossRef] [PubMed]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


Fig. 1 Fig. 2 Fig. 3
Fig. 4

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited