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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Vol. 16, Iss. 2 — Feb. 1, 1999
  • pp: 316–322

Matrix description of near-field diffraction and the fractional Fourier transform

Sara Bradburn Tucker, Jorge Ojeda-Castañeda, and W. Thomas Cathey  »View Author Affiliations


JOSA A, Vol. 16, Issue 2, pp. 316-322 (1999)
http://dx.doi.org/10.1364/JOSAA.16.000316


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Abstract

We describe a matrix multiplication procedure for evaluating the pixelated version of the near-field pattern of a discrete, one- or two-dimensional input. We show that for an input with N×N pixels, in an area d×d, it is necessary to evaluate the Fresnel diffraction pattern at distances zd2/λN. Our numerical algorithm is also useful for evaluating the fractional Fourier transform by multiplying by a special phase matrix with fractional parameter ϵ. If the phase matrix is evaluated at ϵ=1, we find the discrete Fourier transform matrix.

© 1999 Optical Society of America

OCIS Codes
(050.1960) Diffraction and gratings : Diffraction theory
(350.6980) Other areas of optics : Transforms

History
Original Manuscript: May 12, 1998
Revised Manuscript: October 16, 1998
Manuscript Accepted: October 22, 1998
Published: February 1, 1999

Citation
Sara Bradburn Tucker, Jorge Ojeda-Castañeda, and W. Thomas Cathey, "Matrix description of near-field diffraction and the fractional Fourier transform," J. Opt. Soc. Am. A 16, 316-322 (1999)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-16-2-316


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References

  1. W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).
  2. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), pp. 63–105.
  3. V. Arrizón, J. Ojeda-Castañeda, “Fresnel diffraction of substructured gratings: matrix description,” Opt. Lett. 20, 118–120 (1995). [CrossRef] [PubMed]
  4. V. Arrizón, J. G. Ibarra, J. Ojeda-Castañeda, “Matrix formulation of the Fresnel transform of complex transmittance gratings,” J. Opt. Soc. Am. A 13, 2414–2422 (1996). [CrossRef]
  5. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993). [CrossRef]
  6. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980). [CrossRef]
  7. M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representation,” J. Math. Phys. 12, 1772–1780, 1786 (1971). [CrossRef]
  8. R. A. Roberts, C. T. Mullis, Digital Signal Processing (Addison-Wesley, Reading, Pa., 1987), pp. 105–112.
  9. G. B. Parrent, B. J. Thompson, Physical Optics Notebook (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1969), pp. 29–31.

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